Newton to Einstein

Question 1

Define the unit Vector

Answer 1

The unit vector is a vector with a magnitude of one. Any vector is made into its corresponding unit vector by dividing it by its magnitude.

Question 2

Define the Dot Product

Answer 2

C = a . b = |a||b|cos(x) , where x is the angle between a and b when they are ‘tail to tail’

Question 3

Define the cross product

Answer 3

C = a X b = |a||b|sin(x) , where x is the angle from a to b. C is a vector perpendicular to both a and b, its direction depends on the order a and b are crossed.

Question 4

Define the term Kinematics

Answer 4

Kinematics is the study of the relationship between an object’s position, speed and acceleration.

Question 5

Define average velocity

Answer 5

Average velocity is the ratio between the displacement S and the time interval Δt in which the displacement occurs. V = S/Δt

Question 6

Define instantaneous velocity

Answer 6

Instantaneous velocity is the velocity as Δt -> 0. It reaches a limiting value which describes the rate of change of displacement at time t. V = dS/dt

Question 7

Define constant velocity

Answer 7

An object moving with constant velocity will cover equal displacements in equal intervals of time. The magnitude and direction of the velocity will also be constant.

Question 8

Define acceleration

Answer 8

Acceleration is the rate of change of velocity.

Question 9

Express the position of an object as a position vector in terms of its constituent vectors parallel to x and y.

Answer 9

r = xi +yj

Question 10

Express the velocity of an object as a vector in terms of its constituent vectors parallel to x and y.

Answer 10

v ̅=ⅆx/ⅆt i ̂+ⅆy/ⅆt j ̂ = vxi + vyj

Question 11

Express the instantaneous acceleration of an object as a vector in terms of its constituent vectors parallel to x and y.

Answer 11

a = ⅆv/dt = (ⅆv_x)/ⅆt i ̂+(ⅆv_y)/ⅆt j ̂

Question 12

What is the defining characteristic of projectile motion, excluding air resistance.

Answer 12

Acceleration parallel to y only.

Question 13

Define angular displacement

Answer 13

Final Θ - Initial Θ

Question 14

Define angular velocity ω

Answer 14

ω = dΘ/dt
Rate of change of angular displacement

Question 15

Define angular acceleration (alpha)

Answer 15

Rate of change of angular velocity

Question 16

What are the respective signs of angular velocity for acw motion and cw motion

Answer 16

ω > 0 for acw motion , ω < 0 for cw motion

Question 17

What are the defining characteristics of uniform circular motion

Answer 17

ω is constant and therefore ⅆθ/ⅆt is constant. |ω| = 2Π/T where T is the period. However, the direction of motion is constantly changing.

Question 18

Define velocity for uniform circular motion

Answer 18

Because ω is constant, ⅆω/ⅆt = α = 0, the acceleration α is zero tangential to the motion of the particle.
v = ⅆS/ⅆt = ⅆ(rθ)/ⅆt therefore v = r ⅆθ/ⅆt = r ω

v = r ω

Question 19

What Force is responsible for the circular motion of an object.

Answer 19

Centripetal Force given by:
F = mv^2 / r

Question 20

What is the characteristic feature of centripetal acceleration

Answer 20

It is always perpendicular to the velocity of an object exhibiting circular motion

Question 21

Define Centripetal Accelerstion

Answer 21

|a|= ω^2 r = v^2 / r

Question 22

Define the principle of superposition of forces

Answer 22

The net force on an object is the vector sum of the forces acting on it.

Question 23

Give the relationship between angular acceleration and tangential acceleration

Answer 23

alpha = a_t / r

Question 24

Give the rotational kinematic equations

Answer 24

ω_f = ω_i+ αΔt
θ_f = θ_i Δ_t + 1/2 α(Δt)^2
(ω_f)^2 = (ω_i)^2 + 2αΔθ

Question 25

Define Gravitational Force

Answer 25

Gravity: Long range force proportional to the mass. Acts on all objects (moving or at rest) in a gravitational field. F = -GMm/r^2

Question 26

Define spring force

Answer 26

Spring Force: Contact force, either a push or pull. F = -k∆S , where k is the spring constant. The negative indicates that the force acts opposite to the direction of extension.

Question 27

Define the Tension force

Answer 27

Force exerted by a flexible rope/string. Directed along the direction of the rope, away from the object. Always a pull force never a push force.

Question 28

Define the Normal Force

Answer 28

Reaction force that always acts perpendicular to the plane.

Question 29

Define Static Friction

Answer 29

Always acts in the direction that prevents slipping, parallel to the surface. Fs Max = μ_s . n , Where μ_s is the coefficient of static friction. N.B that Fs Max = μ_s . n is not a vector equation. This is because n and Fs are perpendicular to one another.

Question 30

Define Kinetic Friction

Answer 30

Kinetic Friction: Acts opposite to the direction of motion. Fk = μ_k . n

Question 31

Define Drag Force

Answer 31

Acts opposite to the direction of motion of an object in a fluid. D=1/2 CρAv^2 ( (-v ) /|v| )

Question 32

Define Newtons First Law

Answer 32

An object at rest will remain at rest or continue to move with constant velocity in a straight line when Fnet = 0. An inertial reference frame is one in which Newton’s first law is valid (reference frame which is not accelerating)

Question 33

Define Newtons Second Law

Answer 33

An object of mass m subjected to forces F1 F2 F3 etc will undergo acceleration given by a = F/m where F = F1+F2+F3 is the net force.
In general: F=ⅆρ/ⅆt , where p is the change in momentum. ⅆρ/ⅆt = d/dt (mv ) = m ⅆv/ⅆt = ma. Hence force is said to be directly proportional to the rate of change of momentum acting on an object.

Question 34

Define Newtons third law

Answer 34

If object A exerts a force on object B, object B also exerts an equal and opposite force on object A. (action-reaction pair)

Question 35

Define the frictional Force

Answer 35

Friction is a force that opposes the change in motion of an object. The frictional force experienced between two surfaces in contact arises from temporary molecular bonds between the two surfaces.

Question 36

Define Apparent weight

Answer 36

Apparent weight is the magnitude of the contact force supporting an object. It is what a scale would read:
w_app = m(g+a_y)

Question 37

Define Terminal velocity

Answer 37

The velocity at which an object in free fall is no longer accelerating. It is the point where the weight of an object and the drag force on it are equal and opposite.

Question 38

Key Characteristics for Tension in strings and pulleys.

Answer 38

-A string or rope pulls what its connected to with force equal to its tension
-Tension in a rope is equal to the force pulling on the rope.
-Tension in a massless rope is the same at all points in the rope
-Tension does not change when a rope passes over a massless, frictionless pulley.

Question 39

What Force is responsible for a car’s turning circle when travelling round a banked track

Answer 39

For a car travelling round a banked track, static friction is responsible for the cars turning circle as it is always perpendicular to the velocity.

Question 40

Define conservation of Energy

Answer 40

The total energy of the universe is conserved. As the universe is an isolated system, the total energy of an isolated system is also always conserved.

Question 41

Define Mechanical energy

Answer 41

The sum of kinetic and potential energy is called mechanical energy
E_mech = K + U
The potential energy , U, can be any form of potential energy.

Question 42

Define Conservative Forces

Answer 42

A conservative force is a force with the property that the total work done by the force in moving a particle between two points is independent of the path taken.

Question 43

General Strategy for Equilibrium problems

Answer 43

PREP:
Make simplifying assumptions.
Check that object is at rest or moving w/ constant v.
Identify forces.
Draw Free-body diagram.
SOLVE:
Use Newtons 2nd Law in component form.
fx1+fx2+…=0
fy1+fy2+…=0

Question 44

Define Work done by a conservative force

Answer 44

W = -ΔU
Where ΔU is the change in potential energy.

Question 45

Give the relationship between conservative forces and potential energy.

Answer 45

W = -ΔU and W = int[F ds]
Therefore -dU/ds = F

Question 46

General Strategy for Dynamics problems

Answer 46

PREP:
Make simplifying assumptions.
Sketch Diagram.
Identify known quantities.
Identify all forces.
Draw free-body diagram.
SOLVE:
Use Newtons 2nd Law in component form.
fx1+fx2+…=0
fy1+fy2+…=0

Question 47

General Strategy for Problems concerning objects in contact

Answer 47

PREP:
Sketch Diagram.
Identify all forces acting on each object.
Identify action-reaction pairs in the system.
Draw separate free-body diagrams for each object.
SOLVE:
Consider Newtons 2nd Law for each object.
Use Newtons 3rd Law for action-reaction pairs.
Determine how the object’s accelerations are related to each other.

Question 48

Derivation of F=ma

Answer 48

F = dp/dt
dp/dt = d/dt (mv) = m.dv/dt +v.dm/dt = m.dv/dt = ma

Question 49

Derivation of impulse

Answer 49

let J = int[F dt] = int[ma dt] = int[m. dv/dt dt] = mv2-mv1 = Δp
impulse-momentum theorem

Question 50

Define Conservation of momentum

Answer 50

Total momentum of an isolated system is conserved

Question 51

Define Impulse

Answer 51

Impulse if the change in momentum of an object. It is the area under a Force-Time graph.

Question 52

Define Work Done

Answer 52

work done by a force is the dot product of force and displacement
W = F. Δs = FΔscos(x)

Question 53

Define Power

Answer 53

Power is the rate of change of doing work. P = dw/dt = F . V

Question 54

Define a rigid body

Answer 54

Rigid bodies do not change shape in motion these include; translation, rotation and translation and rotation.

Question 55

About what point will a rigid body rotate when not constrained to a fixed axis

Answer 55

its centre of mass

Question 56

How do you calculate centre of mass

Answer 56

x_cm = 1/M int[x dm]
where dm = M.dx / L

Question 57

Define the kinetic energy of a rigid body undergoing rotation

Answer 57

K_rot = 1/2 I ω^2

Question 58

Define moment of inertia

Answer 58

Moment of inertia, I, is a body’s tendency to resist angular acceleration, which is the sum of the products of the mass of each particle in the body with the square of its distance from the axis of rotation. It is rotationally analogous to mass.
I = mr^2

Question 59

Give the expression for the kinetic energy of a body exhibiting translational and rotational motion

Answer 59

K = 1/2mv^2 + 1/2 I ω^2
Sum of translational kinetic and rotational energy

Question 60

How do you find the moment of inertia of an extended object.

Answer 60

Sum over all the moments of inertia of constituent points of the extended object

Question 61

Define rolling without slipping

Answer 61

If a wheel is not slipping with respect to the ground, then the point is, at that instant, at rest relative to the ground. This type of motion is “rolling without slipping”; the point on the rotating object that is in contact with the ground is instantaneously at rest relative to the ground.
Mechanical energy is conserved.

Question 62

What is the perpendicular axis theorem

Answer 62

For a planar object: If I_x and I_y are the moments of inertia about two perpendicular axis x,y, in the plane of the object, which meet at the origin, then the moment of inertia about an axis z through the origin is given by:
I_z = I_x + I_y

Question 63

What is the parallel axis theorem

Answer 63

If the moment of inertia of a body of mass M about an axis through its centre of mass is I_cm, then the moment of inertia about an axis parallel to this but a distance d from it is given by: I = I_cm + Md^2

Question 64

What Three variables does the ability of a force to cause rotational motion depend on

Answer 64

1) The distance ‘r’ from the rotational axis
2) The magnitude of the applied force
3) The angle of the applied force relative to the radial line

Question 65

Define Torque

Answer 65

Torque is rotationally analogous to Force and is given by:
τ = r F sinφ (Nm)
where φ is the angle between the force and radial line.
Torque is +ve when the consequent rotation is anticlockwise and -ve when clockwise.
Hence torque can be expressed as the cross product of the r positional vector and Force vector such that:
Τ = r X F
The direction of the torque vector must be perpendicular to both the radial line and force vector due to the nature of the cross product.

Question 66

Define net torque on an object

Answer 66

τ_net = I α
where I is moment of inertia and α is angular acceleration.
It is the rate of change of angular momentum.

Question 67

Define Static Equilibrium

Answer 67

When referring to an extended object, static equilibrium is such that there is no linear or angular acceleration and thus the net force and torque are both zero

Question 68

What rule determines the direction of the angular velocity vector ω

Answer 68

Right hand rule. Thumb indicates direction of ω, Fingers indicate direction of rotation.

Question 69

Define Angular momentum of a point particle about the origin

Answer 69

L = r X p = mrvsinφ
= m(r X v)
Where L is the angular momentum, p is linear momentum and r is the positional vector from the origin.

Question 70

Define conservation of momentum

Answer 70

The angular momentum of the system is conserved.
In an isolated system angular momentum is always conserved and hence the net angular momentum before an event must equal the net angular momentum after an event.

Question 71

What is the angular momentum for a particle exhibiting circular motion

Answer 71

L=mvr, this is because v and r are always tangential.

Question 72

Explain the procession of a gyroscope

Answer 72

In the same way Force can change the momentum of an object, torque can change the angular momentum of an object:

When the gyroscopic wheel is not spinning, gravity creates a torque that is in the plane of the wheel. This results in a rotation downwards causing it to fall

When the wheel is spun an angular momentum is created that comes out the plane of the wheel away from the pivot. Because gravity exerts a constant force on the gyroscope, there will be a constant torque. As the angular momentum changes, so does the direction of the torque. The torque and angular momentum vectors are therefore always at 90 degrees creating a processional motion

Question 73

Define angular momentum for an extended object rotating about one of its axis

Answer 73

Angular momentum is given by:
L = I ω.

Question 74

Define Oscillatory Motion

Answer 74

Repetitive motion about an equilibrium position. It can be periodic or non periodic

Question 75

What is the defining characteristic of simple harmonic motion

Answer 75

For an oscillation to exhibit SHM the acceleration of the system must be opposite and proportional to the displacement.

Question 76

Give the equations of displacement, velocity and acceleration for mass on a spring subject to x = A, at t = 0 with V = 0

Answer 76

X = A(cos 2πt/T) = Acos(2πft) = Acos(wt)

V = d/dt [Acos(wt)] = -Awsin(wt)

A = d/dt [-Awsin(wt)] = -A w^2 cos(wt) = -w^2 x

Question 77

Give the disaplacement equation for a mass on a spring that doesnt start from x=A at t=0

Answer 77

X = Acos(wt + φ)
-π < φ ≤ π
In this case, a phase constant is introduced such that the initial conditions of the system are represented in the equation

Question 78

Give the conditions for which X = Acos(wt + φ) is a solution to the differential equation (ⅆ^2 x)/(ⅆt^2 )+ K/m x = 0

Answer 78

w^2 = k/m and T = 2π√(m/k)

Question 79

Give the form of the solution and the physical definitions for the solutions of differential equations corresponding to damped oscillators

Answer 79

The solution cannot be obtained by direct integrations. Therefore we consider a possible solution x = Ae^αt
x = Ae^αt , x’ = αAe^αt , x’’ = α^2Aeαt
Substitute into the differential equation: α^2Aeαt + b/m αAeαt + k/m Aeαt = 0
This creates an auxiliary equation: α^2 + b/m α + k/m = 0
Hence Ae^αt Is a valid solution if α satisfies the auxiliary equation.
The discriminant of the Auxiliary equation determines the final form of the solution to the differential equation
Light damping: discriminant is negative. Two values for alpha that are both complex
Heavy damping: discriminant is positive. Two real values for alpha
Critical damping: discriminant is zero. One real value for alpha

Question 80

Define a travelling wave

Answer 80

An organised disturbance that propagates through space with a well-defined wave speed. Waves transfer energy but do not transfer matter.

Question 81

Define a mechanical wave

Answer 81

Mechanical waves are waves that involve the motion of substance through which they move, the medium.
As a wave passes through a medium, the elements that make up the medium are displaced from their equilibrium position. This is a disturbance of the medium.

Question 82

Give the defining characteristics of transverse and longitudinal waves respectively

Answer 82

Transverse waves oscillate perpendicular to the propagation and longitudinal waves oscillate parallel to the propagation.

Question 83

Give the wavespeed equation

Answer 83

v=fλ
the wave speed is also described by
v= ω/k

Question 84

Give the wavespeed for waves on a string

Answer 84

v= √(T/µ) where T and µ is the tension in the string and the linear density of the string respectively.

Question 85

What do snapshot graphs of a wave show.

Answer 85

Oscillatory displacement against linear displacement for a given time

Question 86

What do history graphs of a wave show

Answer 86

Oscillatory displacement against time for a given point in space

Question 87

What type of source gives a sinusoidal wave

Answer 87

Wave sources that oscillate with simple harmonic motion generate sinusoidal waves. The frequency of the wave is the frequency of the oscillating source.

Question 88

Define the wavenumber

Answer 88

wave number k= 2π/λ

Question 89

define the angular frequency

Answer 89

angular frequency ω= 2π/T

Question 90

Give D(x,t) for a sinusoidal wave travelling in the positive and negative directions respectively.

Answer 90

D(x,t) = Asin(kx-ωt+φ) +ve

D(x,t) = Asin(kx+ωt+φ) -ve

Question 91

For a wave travelling on a string where the displacement is a function of position and time given by;
y(x,t) = Asin(kx-ωt+φ)
What is the velocity of the string along the y direction?

Answer 91

The partial derivative of y with respect to t gives the velocity of the string along y;
(∂y(x,t))/∂t= -ωAcos(kx-ωt+φ)

Question 92

For a wave travelling on a string where the displacement is a function of position and time given by;
y(x,t) = Asin(kx-ωt+φ)
What is the slope of the string at a given time?

Answer 92

Partial derivative of y with respect to x gives the slope of the string at a given time t;
(∂y(x,t))/∂x= kAcos(kx-ωt+φ)

Question 93

Describe fixed end reflection

Answer 93

Fixed end reflection:
The incident pulse is reflected at the pole. and the reflected pulse is inverted.
The upward-pull (action) from the string on the pole does not change the pole, which is fixed. The downward pull (reaction) from the pole on the string results in the upward displacement to become a downward displacement.

Question 94

Describe free end reflection

Answer 94

The incident pulse is reflected at the pole and the reflected pulse is not inverted. When the incident pulse reaches the end of the medium, the string does not interact with the pole.

Question 95

Describe what happens when a pulse crosses a boundary from a less to more dense string.

Answer 95

Upon reaching the boundary, the pulse is reflected and inverted.

A portion of the energy carried by the incident pulse is transmitted into the thick rope. The disturbance that continues moving to the right is the transmitted pulse (which is not inverted). The reflected pulse in this situation will not be inverted. Similarly, the transmitted pulse is not inverted (as is always the case). Since the incident pulse is in a heavier medium, when it reaches the boundary, the first particle of the less dense medium does not have sufficient mass to overpower the last particle of the more dense medium.

The result is that an upward displaced pulse incident towards the boundary will reflect as an upward displaced pulse.

Question 96

Give the principle of superposition

Answer 96

Principle of Superposition: When two or more waves are simultaneously present at a single point in space, the displacement of the medium at that point is the vector sum of the individual displacements due to the individual waves.

Question 97

Define interference

Answer 97

The superposition of two waves, and their resultant sum, is known as interference. When the individual displacements at a point are both positive, the amplitude of the resultant displacement is bigger than either of the constituent displacements, this is constructive interference. If one of the individual waves has negative displacement, the resultant displacement is smaller than either one of the constituent displacements.

Question 98

What leads to a standing wave

Answer 98

The superposition of two sinusoidal waves travelling in opposite directions, but with the same frequency, wavelength and amplitude leads to a standing wave.

Question 99

Describe a standing wave

Answer 99

A standing wave is one with fixed nodes and antinodes. The amplitude of a standing wave is a function of the relative displacement of the point in the medium. i.e different points have fixed maximum amplitudes and oscillate with SHM. Nodes have a fixed amplitude of zero, antinodes will oscillate with the maximum amplitude of the entire wave.

Question 100

Give the equation for a standing wave

Answer 100

D(x,t) = 2asin(kx)cos(wt) = A(x)cos(wt)

Each point along the wave oscillates with SHM with an amplitude A(x)=2asinkx that depends on position x.
A(x) has maximum/minimum positions called antinodes separated by half a wavelength.

Question 101

Give the boundary conditions and their results for a standing wave on a string.

Answer 101

The boundary conditions are set by the fact the string is fixed at both ends, therefore both ends of the string are nodes.

D(0,t)=2a sin(k0)cos⁡(ωt)=0 always satisfied

D(L,t)=2a sin(kL)cos(ωt)=0 satisfied if sinkL=0

Boundary condition at x=L requires that kL= 2πL/λ
=mπ
where m is a positive integer.

λ_m = 2L/m and f_m = mv/2L

Question 102

What are possible standing waves called that meet the boundary conditions

Answer 102

Possible standing waves that meet the boundary conditions are called normal modes of the string.

Question 103

How are sound waves created from a loudspeaker

Answer 103

As the loudspeaker cone oscillates back and forth it creates regions of higher and lower air pressure.
The disturbance D(x, t) can be described in terms of - the displacement Δx(x,t) of the molecules from equilibrium. - the change of pressure p(x,t) from an equilibrium pressure.

Question 104

How does the speed of sound depend on the properties of the medium

Answer 104

The speed of sound waves in a medium depends on the properties of the medium; mass density and compressibility.

The relationship between the speed, density and compressibility s given by; v= √(1/pk)

Where p is the density p=m/v
k is the compressibility given by k= (-1)/V dV/dP . This is the relative change of volume dv/v in response to a pressure change dP.

Question 105

Describe the helium effect on the voice

Answer 105

The human voice originates when the air flowing up the trachea undergoes pressure modulations as it passes between the vibrating vocal chords in the larynx. The sound produced consists of a fundamental frequency, which determines the voice’s pitch, and harmonics (integral multiples) of this frequency. The average frequency of the fundamental is ~ 100-200 Hz. Sound waves bouncing back and forth within the cavity superimpose to produce a loud sound with the lowest-frequency peak called the fundamental. The fundamental frequency of a resonating cavity is directly proportional to the speed of sound in the gas occupying the cavity. The resonance frequencies of the vocal tract are ~ 3 times higher for helium than for air. At a pressure of one atmosphere, with pure helium in your vocal tract instead of air, the pitch of your voice will be about 3 octaves higher than usual (like Donald Duck’s).

Question 106

Give the equation of superposition of two sinusoidal waves given by
D_1 (x_1,t)=asin⁡(kx_1-ωt+φ_10 )
D_2 (x_2,t)=asin⁡(kx_2-ωt+φ_20)

Answer 106

D = D1+D2=asin⁡(kx_1-ωt+φ_10 )+asin⁡(kx_2-ωt+φ_20 )=asinφ_1+asinφ_2
Using sin⁡α±sin⁡β=2 sin⁡〖1/2 (α±β)〗 cos⁡〖1/2 (α∓β)〗
D=2 acos⁡((φ_2-φ_1)/2)sin⁡((φ_1+φ_2)/2)
D=2 acos⁡(Δφ/2)sin⁡[(k(x_1+x_2 )/2)-ωt+⁡((φ_10+φ_20)/2)]
The amplitude of the wave is A=2 acos⁡(Δφ/2) and depends on the phase difference.

Δφ = φ_2-φ_1 is the phase difference between the two waves
x_avg=(x_1+x_2)/2 is the average distance to the two sources
φ_0avg=(φ_10+φ_20)/2 is the average phase constant of the two sources
This gives; D=2 acos⁡(Δφ/2)sin⁡[kx_avg-ωt+⁡φ_avg]

Question 107

Give the conditions for maximum constructive interference for two identical sources.

Answer 107

Δφ = φ_2-φ_1=m2π
Δφ = φ_2-φ_1=kx_2-ωt+φ_2-kx_1+ωt-φ_10=kΔx+Δφ_0
Where the path length difference is Δx=x_2-x_1
The inherent phase difference is Δφ_0=φ_20-φ_10
Thus the condition for maximum constructive interference can be written as;
Δφ=2π Δx/λ+Δφ_0=m2π where m=0,1,2…
For identical sources Δφ_0=0 hence Δφ=2π Δx/λ=m2π such that Δx=mλ
Hence, maximum constructive interference occurs when the sources are separated by a multiple of the wavelength

Question 108

Give the conditions for maximum destructive interference for two identical sources.

Answer 108

Δφ=2π Δx/λ+Δφ_0=(m+1/2)2π where m=0,1,2…
For identical sources Δφ_0=0 hence Δφ=2π Δx/λ=(m+1/2)2π such that Δx=(m+1/2)λ
Hence, maximum constructive interference occurs when the sources are separated by a half integer multiple of the wavelength.

Question 109

Define the doppler effect

Answer 109

The doppler effect is the change in frequency of a wave for an observer moving relative to the source of the wave.
Consider a source that generates sinusoidal waves with frequency f0 , an observer receding from the source detects f-<f0 and an observer approaching the source detects f+>f0 .

Question 110

Give the frequency detected by a moving observer for a stationary source

Answer 110

Moving Observer, Stationary source

In the reference frame of the observer the wave travels with speed v’ = v0 + v
The frequency detected by the observer is given by; f_±=f_0 (1±v_0/v)
V0 is the speed of the observer and V is the wave speed.

f_+=f_0 (1+v_0/v) Observer approaching source
f_-=f_0 (1-v_0/v) Observer receding from source

Question 111

Give the frequency detected by a stationary observer for a moving source

Answer 111

Stationary Observer, Moving Source

The frequency detected by the observer is given by; f_±=f_0/(1 ∓ v_s/v)
Vs is the speed of the source and V is the wave speed.

f_+=f_0/(1 - v_s/v) Source approaching observer
f_-=f_0/(1 + v_s/v) Source receding from observer

Question 112

Define the reference frame

Answer 112

Define a reference frame to be a coordinate system in which experimenters equipped with meter sticks, stopwatches and other necessary equipment make position and time measurements on moving objects.
Three implicit ideas are defined by the reference frame;
-The reference frame extends infinitely far in all directions
-Experimenters are at rest in the reference frame
-The measurements taken in a reference frame are accurate to any level of accuracy needed

Question 113

Define an inertial reference frame

Answer 113

We define an inertial reference frame as one in which Newton’s first law is valid. This means that objects in this inertial reference frame are stationary or moving with constant velocity when the resultant force upon them is zero. In general non-inertial reference frames are ones that are accelerating. The relative velocity between two inertial reference frames is always constant.

Question 114

Define an event

Answer 114

Events are fundamental in relativity Define an event as a physical activity that takes place at a definite point in space and at a definite instant in time.

Where and when an event occurs can be measured in different reference frames. For the same event, coordinates may differ between reference frames.

Question 115

What is Einstein’s basic postulate of relativity and its implications

Answer 115

Einstein proposed that all the laws of physics are the same in all inertial reference frames, this is known as the basic postulate of relativity. This statement implies that maxwell’s equations are laws of physics that hold true in all inertial reference frames and consequently the speed of light is invariant in all inertial reference frames.
This also implies that two events that are simultaneous in one frame of reference may not be simultaneous in another.

Question 116

Define the relativity of simultaneity

Answer 116

Two events that take place at different positions but at the same time, as measured in a given reference frame, are said to be simultaneous in that reference frame.
Simultaneity is determined by when the events actually happened and not the time of any one observation.

Question 117

What is a light clock.

Answer 117

The light clock is a box of height h with a light source at the bottom and mirror at the top. The light source emits a very short pulse of light that travels to the mirror and reflects back to the detector beside the source. A ‘tick’ is equivalent to each time the detector receives a light pulse, it then immediately emits the next pulse.
By comparing time intervals, ‘tick’, for a light clock in a rest frame and a moving frame, the principle of time dilation can be analytically derived.

Question 118

Define proper time

Answer 118

The time interval between two events that occur at the same position is called proper time Δτ. In the light clock example, the proper time is the time interval measured in the rest frame, this is because there is no displacement between events.

Question 119

Define the proper length

Answer 119

The distance L between two objects, or two points on one object, measured in the reference frame where the objects are at rest is called the proper length, L .

Question 120

Define the spacetime interval

Answer 120

We can define the spacetime interval s^2= c^2 Δt^2 - Δx^2.
All observers will agree on the value s. The spacetime interval between two events in invariant in all inertial reference frames.

Question 121

Define relativistic momentum

Answer 121

We define relativistic momentum using proper time; p=m Δx/Δτ , using Δt=Δτ/√(1-B^2 ) it can be written that; p=γmv
Where m is the rest mass of the particle.

Question 122

give the energy-momentum invariant expression

Answer 122

E^2-(pc)^2=(mc^2 )^2=〖E_0〗^2

Question 123

Continuous spectra

Answer 123

Continuous spectra; We can measure intensity (brightness) of emitted light at each wavelength 𝜆 * At each 𝜆, intensity increases with temperature 𝑇 * Wavelength at peak of intensity 𝜆peak ∝ 1/𝑇 * Many materials have intensity curves very similar to the plotted ones, called the blackbody spectrum. All other materials’ spectra fall below these curves (i.e., intensity is lower for each 𝜆 and 𝑇)

Question 124

Discrete spectra

Answer 124

Discrete Spectra; A diffraction grating can also be used to measure the absorption spectrum of a gas. Sodium vapour Gases also absorb only certain discrete wavelengths: … which are a subset of the emission wavelengths.

Question 125

Oil drop experiment

Answer 125

• “Atomizer” produces oil drops of size ∼ 1 μm which fall due to gravity
• Some drops are charged by friction in sprayer
• Charged drops can be suspended by 𝐸 field between parallel-plate electrodes.
  In equilibrium, Fgravity + F_E = 0
  mdropg = qdropE

To find mdrop turn off the E and measure terminal velocity of falling drop
Radius of drop; Fdrag = k r_drop v (stoke’s law)
Mass of drop; mdrop = p_oilX 4/3pi r^3_drop
Charge of drop; qdrop = mdropg/E
All drops have charges that are integer multiples of a certain base value: |qdrop| = N X e where e = 1.6X10-19
Hence charge is quantised.

Question 126

Give the results of Phillipp lenards experiment

Answer 126

1 -Current is proportional to light intensity
2 -Current flows with no delay
3 -Current only if light frequency f>f¬0 threshold frequency
4 -f0 is a property of the cathode material
5 -Current depends on p.d, I increases linearly for V>0,
I=0 for V<-Vstop , Vstop is ‘stopping potential’
6 -Vstop is Independent of light intensity.

Question 127

Give the classical interpretation of lennards results

Answer 127

The classical interpretation explains some but not all of Lenard’s results;
Light inputs energy to material, heating it up and causing thermal emission.
Every material has a work function E0, the minimum energy needed to liberate an electron from within it.
¬¬K¬max¬ = Eelec – E0 for emitted electron leaving cathode. Max KE at anode is Kf = Kmax + eV.
If V<0, electrons are decelerated. If V < -K¬max/e , even fastest electrons wont make it to the anode and therefore there will
be no current. Vstop = Kmax/e

Question 128

What is debroglies proposition and equation

Answer 128

Recall E^2-(pc)^2=(mc^2 )^2
For photons m=0 hence E^2=(pc)^2 so E=pc=hc/λ and λ=h/p (Debroglie Equation)

In 1924 de Broglie proposed that any particle can behave as a wave with wavelength given by the de broglie equation.

Question 129

What is the general result for a particle of mass m confined in a region of length L (particle in a box)

Answer 129

General result; For a particle of mass 𝑚 confined in a region of size 𝐿 * Energy is quantized: only discrete values of energy are possible * There is a minimum possible energy 𝐸1 * Both the separations Δ𝐸 and the minimum 𝐸1 are ~ h^2/mL^2

Question 130

Give Bohr’s model of quantisation

Answer 130

An atom has a discrete set of stationary states, each with a well-defined energy En.
The states are labelled by their principal quantum number, n=1,2,3… in order of increasing energy.
Each state can be imagined as having the electrons in a certain set of discrete orbits (“shells”) around the nucleus.

Question 131

Give the classical equation for an electron in orbit about the nucleus

Answer 131

(mv^2)/r=e^2/(4πε_0 r^2 )

Question 132

Derive the angular momentum of an electrons orbit

Answer 132

A whole number of wavelengths must fit in the circumference of the orbit, so 𝑛𝜆 = 2𝜋r

From de Broglie’s equation λ=h/p =h/mv therefore nh/mv=2πr for n = 1,2,3…
This gives mvr=nℏ where ℏ=h/2π
This means that L=nℏ =mvr where L is the electrons orbital angular momentum.
Hence L is always an integer multiple of h-bar, meaning h-bar is the fundamental unit of angular momentum.

Question 133

Derive the bohr radius

Answer 133

We have two equations in the two unknowns v and r.
Classical orbits; (mv^2)/r=e^2/(4πε_0 r^2 )
Circular de Broglie standing waves; nh/mv=2πr
Rearranging gives r_n=(4πε_0 r^2 ℏ^2)/(me^2 ) n^2=a_B n^2 v_n= e^2/(2πε_0 hn)=ℏ/(ma_B n)
Where a_B=(4πε_0 r^2 ℏ^2)/(me^2 )=0.0529nm is the Bohr radius.

Question 134

Deduce the born rule using a double slit experiment of monochromatic light

Answer 134

Light waves from the two slits interfere to give a standing wave at the screen with amplitude function A(x). The amplitude function has peaks where the two constituent waves peaks overlap and troughs where constituent troughs overlap w/ nodes in between.
Intensity I(x) is energy per second per unit area at position x. I(x) ∝ [A(x)]^2 so intensity maxima are at both peaks and trough of A(x) and minima are at nodes.
Light is made of photons and each has same energy given by E=hf so rate of photon arrival at x is R(x) ∝ I(x) ∝ [A(x)]^2 . But photons move independently and arrive one by one, each at a given position. Therefore each photon must have a probability of arriving at x that is proportional to [A(x)]^2 , the square of the wave amplitude.
Position takes continuous values so its distribution is described by a probability density function.

Define probability density P(x).
Probability of arrival within a segment length δx; P(in δx at x)=P(x)δx.
Total probability of arriving between x=a and x=b is P(a<X<b)=∫P(x)dx between a and b.

• Electrons show the same type of interference pattern
• Each electron must also have a probability density function 𝑃(𝑥) for the arrival position 𝑥
• For photons, we have P(x)∝[A(x)^2
• For electrons and other particles, we define a function 𝜓 (𝑥) so that P(x)=[ψ(x)]^2.
  This is the Born rule.

Question 135

What is the normalisation condition of the wavefunction

Answer 135

For a continuous distribution with probability density function P(x):
Probability of arrival within a segment length δx; P(in δx at x)=P(x)δx
Total probability of arriving between x=a and x=b is P(a<X<b)=∫P(x)dx
Since the particle must be somewhere P(-∞<X<∞)=∫P(x)dx between -ve and +ve infinity =∫[ψ(x)]^2 dx = 1
This is called the normalisation condition on ψ(x).

Question 136

Define a wave packet

Answer 136

in quantum mechanics, a wave packet is a localised oscillation analogous to a classical particle.

Question 137

Explain the Heisenberg Uncertainty Principle

Answer 137

The wave packet is spread out over distance Δx. Its position is uncertain.

The wave packet is constructed from a range of wave vectors Δk. Momentum p = h/λ = hk/2π, so the wave packet contains a range of momenta:
Δp = ℏΔk. Its momentum is uncertain

Combining Δp = ℏΔk and ΔxΔk≈2π gives :
Δ𝑥 × Δ𝑝 ≳ ℎ

This is the Heisenberg uncertainty principle.

“≳” means “greater than a value approximately equal to”
* “greater than” because this is the “best case scenario”: there can always be more uncertainty in practice.
* “approximately” because we did not define Δ𝑥 precisely; some people (e.g., Knight) write ℎ/2 instead.

Question 138

Compare Classical and Quantum stationary states

Answer 138

Classically:
-Particle can perform periodic oscillations with any energy.
-At each point in time the particle is at a single point in the oscillation.

QM:
-Discrete set of stationary states, each with a well defined energy and wave function.
-Particle is everywhere in the oscillation at once with a certain time independent probability to be found in any position.

Question 139

Give the conditions on physical solutions of the shrodinger equation.

Answer 139

1. 𝜓 (𝑥) must be a continuous function – no jumps.
2. The derivative of 𝜓 (𝑥) must be continuous except where U(x) is infinite
3. 𝜓 (𝑥) = 0 where U(X) is infinite (forbidden regions)
4. ∫(-∞)^∞▒〖P(x)dx=∫(-∞)^∞▒〖|ψ(x)|^2 dx〗〗=1
5. 𝜓 (𝑥) must go to zero for x = ±∞.

These conditions lead to the quantisation of energy, i.e all solutions of the schrodinger equation only exist for discrete energy values.

Question 140

Define the classically forbidden region

Answer 140

Classically, the kinetic energy is a positive number thus the classically forbidden region is where the kinetic energy would be negative, or in other words where the potential energy is greater than the total energy.

Question 141

Describe the particle in a box model

Answer 141

We can model trapping with a “well” potential 𝑈(𝑥), low inside the well and high outside. A particle with total energy 𝐸 less than the top of the well is trapped inside.
We make three simplifications: 1. constant (flat) 𝑈 (𝑥) inside and outside 2. vertical “walls” 3. 𝑈 (𝑥) → ∞ outside (so any 𝐸 is trapped).
This is called in an infinite square well or a box

Question 142

Describe the finite potential well

Answer 142

Defined by zero potential in some given interval with high but finite potential elsewhere.

Question 143

What does the penetration describe about the wave function

Answer 143

Penetration distance η= ℏ/√(2m[U_0 - E] ) gives the length scale over which ψ (x) extends outside the well ’boundary’.
For example at x=L+η , ψ (L+η) = ψ_edge.e^(-1)=0.37ψ_edge

Question 144

What is the quantum correspondence principle

Answer 144

We know that Newtonian physics correctly describes the everyday world. So our new theories should agree with Newton where Newton’s laws apply: for “big”, “slow” things. This is called the “Correspondence Principle”.
Quantum states with large 𝑛 (large objects) should be approximately classical, 𝑃quant (𝑥) ≈ 𝑃class (𝑥) , at least on average.

Question 145

What are the General principles to sketch wave functions

Answer 145

• 𝜓 (𝑥) must be continuous; so must its derivative, except at 𝑈 (𝑥) = ∞, where 𝜓 (𝑥) = 0 [SP7].
• 𝜓 (𝑥) decays exponentially where 𝐾 = 𝐸 − 𝑈 𝑥 < 0 (classically forbidden regions), with penetration distance;
  η∝1/√(U (x) - E) [SP9].
• 𝜓 (𝑥) oscillates where 𝐾 = 𝐸 – 𝑈(𝑥) > 0. Where 𝐾 is smaller, the wavelength is longer (because 𝜆 = ℎ/𝑝) and the oscillation amplitude is larger (correspondence principle).
• Node theorem: State 𝑛 has 𝑛 − 1 nodes (crossings through 𝜓 = 0) excluding the ends (and so usually 𝑛 antinodes).

Question 146

What is the law of conservation of charge

Answer 146

Law of conservation of charge; charge is neither created or destroyed, it can only be transferred. The net charge of a given system is constant.

Question 147

Properties of Insulators

Answer 147

*Electrons tightly bound to the positive nuclei and not free to move around.
*Charging an insulator by friction leaves patches of molecular ions on the surface, but these patches are immobile.
*Plastic is an insulator

Question 148

Properties of conductors

Answer 148

*In metals, outer atomic electrons weakly bound to the nuclei and become detached from their nuclei.
*Electrons are free to move through the entire solid, acting like a negatively-charged liquid permeating an array of positively charged ion cores.
*Metals are good at conducting electricity
*The solid as a whole remains electrically neutral.

Question 149

Describe electric dipoles

Answer 149

If an external electrical force is applied to an atom, an electric dipole is induced.
* With external charge, electron cloud (or electron sea for metals) is displaced ⇒ polarization
* Net attractive force.
* Two opposite charges with a slight separation between them form an Electric Dipole.
The dipole is the partial charge separation within the atom.

Question 150

Describe coulombs law

Answer 150

If two charged particles (point charges) having charges q1 and q2 are at a distance r apart, the particles exert forces on each other of magnitude;
*These forces are an action-reaction pair, equal in magnitude and opposite in direction. *The forces are directed along the line joining the two particles.
*The forces are repulsive for two like charges and attractive for two opposite charges.
*The unit of charge is the coulomb [C]

Electric Force is a vector. The principle of superposition for electric forces states that for a given test charge, the net electric force acting on the test charge is the linear vector sum of all the electric forces acting on that test charge.

Question 151

A charge q at a point in space where the electric field is 𝐸 experiences an electric force give by…

Answer 151

F=qE

Question 152

Define the electric field model

Answer 152

The electric field at some point is defined as the ratio between the force a test charge q experiences as a result of the field and the magnitude of q.

Question 153

Give the properties of electric field lines

Answer 153

*Electric field lines are continuous curves tangent to the electric field vectors
*Closely-spaced lines represent large field strength
*Electric field lines never cross
*Electric field lines start on positive charges and end in negative ones.

Question 154

Define linear charge density

Answer 154

The linear charge density of an object of length L and charge Q, is defined as
λ = Q/L
Linear charge density, which has units of C/m, is the amount of charge per meter of length.
Assumption: object is uniformly charged, otherwise 𝜆 = 𝜆(𝑥).

Question 155

Define symmetry in the context of the E-Field of an extended object

Answer 155

A system is symmetric when there is a group of geometric transformations that do not cause any physical change to the system.

The symmetry of the E-field of an extended, charged object must match the symmetry of the charge distribution.
For a uniformly charged object;
If a geometrical transformation doesn’t affect the charge distribution but affects the electric vector field due to that charge distribution, then the electric field is invalid and cannot exist.

Question 156

Define electric flux

Answer 156

The ‘amount’ of Efield passing through a surface is called the electric flux.
Electric flux = φ
φ=EAcosx=E∙A (N/C)∙m^2
The area vector A is perpendicular to the surface and its magnitude is the area of the surface.

Question 157

Give the properties of conductors in electrostatic equilibrium

Answer 157

The electric field inside has to be zero: 𝐸 = 0
If this weren’t true, there would be forces 𝐹 = 𝑞𝐸 on the charges and they would move.
For any gaussian surface inside the conductor E=0, phi=0, Qin=0 therefore All the excess charge of a charged conductor resides on the exterior surface of the conductor.

The electric field on the surface has to be has to be perpendicular to the surface. IF this weren’t true, there would be components tangential to the surface which would exert a force F=qE on the charges and a surface current would be induced.

Question 158

Define the electric field

Answer 158

Possible to define an electric potential U_elec as the work needed to move a test charge q through an electric field E from one point to another.

The electric field does work on the particle, this work can be expressed as a change in electric potential energy

ΔUelec = Uf – Ui = -W_elec

For a positive change, the field direction is downhill, U_elec decreases as the charge speeds up and vice versa.

The potential energy change of a test charge q in a uniform electric field is given by;
W_elec = qE|f – Si|

Question 159

Define the electric potential

Answer 159

Electric potential is defined as the amount of work energy needed per unit of electric charge to move the charge from a reference point to a specific point in an electric field.
The electric potential is defined as V≡U_(q+sources)/q
Charge q is used as a probe to determine the electric potential, but the value of V is independent of q. The electric potential, like the electric field, is a property of the source charges.

Question 160

How do you find the potential from the electric field

Answer 160

1. Draw a diagram and identify the point at which you wish to find the potential. Call this position i.
2. Chose the zero-point of the potential, often at infinity. Call this position f.
3. Establish a coordinate axis from i to f along which you already know or can easily determine the electric field component Es.
4. Integrate the following equation to find the potential: -int[E.ds]

Question 161

Define an equipotential

Answer 161

When a charge is moved along an equipotential line its potential energy does not change. This is because The equipotential surface is always perpendicular to the electric field.
The electric field points in the direction of decreasing potential.

Question 162

Explain why charge density is higher at points with a high degree of curvature for extended objects.

Answer 162

• Entire conductor has the same potential. ⇒ surface is an equipotential surface ⇒ 𝐸 is perpendicular to the surface.
  When an electron is close to the edge of the conductor’s surface, there is a net restoring force that keeps the electron bound within the conductor. This is a result of the large potential difference between the lattice and the region outside, this difference is the work function of the electron.
  For a pair of electrons interacting near the surface with minimal curvature, coulomb’s law dictates the charge density. Near the surface with a high degree of curvature (corners), the effectiveness of coulomb’s law is reduced by the component of the large restorative force that keeps the electron bound in the conductor lattice. Hence electron repulsion is reduced at corners of the conductor and charge density is greater.

Question 163

Give the potential difference and capacitance across a capacitor

Answer 163

〖ΔV〗_c= Ed , E =Q/εA hence Q =(εAΔV_c)/d
Define capacitance for a parallel-plate capacitor C=Q/(ΔV_c )=(ε_0 A)/d

Question 164

Energy stored in capacitor

Answer 164

U_c=1/2 C(ΔV)^2=1/2 (ε_0 A)/d(Ed)^2=1/2 εAdE^2

Question 165

Energy density of capacitor

Answer 165

U_E = U_c / volume = U_c/ Ad

Question 166

Describe what happens to the charges in a conductor when a conductor in inside a non zero E field

Answer 166

If we put a conductor into a region with non-zero electric field (e.g. a conducting sphere in between the plates of a capacitor). Charges move to surface to screen the external field – new equilibrium.

Question 167

Describe electric current density

Answer 167

Electric current density, j , defined at point in space in terms of rate at which charge would cross a surface element.
Consider a small element of area dS – describe with a vector, (dS) , where |(dS) | is the area of the element.
Vector (dS) points perpendicular to area element.
Net charge that passes through element dS in time δt is given by; δq = j ∙ (dS) δt

Question 168

Describe the relationship between electric current density and the electric field weakly out of equilibrium

Answer 168

Weakly out of equilibrium, the relationship between j and E is linear:
j =σE and E =ρj
where ρ=1/σ
ρ=resitivity
σ=conductivity

Question 169

Define kirchoffs loops law

Answer 169

The work done by the electric forces is independent of the path followed ⇒ the potential difference between two points must also be independent of the path followed ⇒ the sum of all the potential differences encountered while moving around a loop or closed path is zero.  Kirchhoff’s loop law.

Question 170

Define kirchoffs junction law

Answer 170

Law of Conservation of Current: The current is the same in all points in a current-carrying wire. This is a consequence of the law of conservation of charge (i.e. the number of electrons).
⇒ The sum of the currents into a junction must equal the sum of the currents leaving the junction. Kirchoff’s junction law.

Question 171

Drude Model

Answer 171

• Quasi-free “charge carriers” in a conducting solid
• Carriers have charge, q, and mass, m.
• Carriers feel force 𝐹 = 𝑞𝐸 due to electric field – acceleration qE/m.
• Carriers undergo frequent random scattering events (“collisions”)
• Scattering event randomizes the direction of a carrier’s velocity (may also redistribute energy to other parts of the system)
• Scattering processes lead to a frictional drag force that balances electric field leading to a constant average drift velocity for the charge carriers.

Question 172

Problems with the drude model

Answer 172

• Different experiments to measure carrier mass, m, give different results
• Hall effect experiments measure charge on carriers – in some materials, carriers are positively charged
• Quantum mechanics – electrons are quantum particles. Quantum treatment of particles in a perfect periodic potential shows that such a potential does not scatter electrons
• Carriers are not free electrons that scatter off lattice of ions

Question 173

Give the 5 basic experimental magnetic results

Answer 173

1.Magnetism is not the same as electricity.
2.Magnetism is a long-range force.
3.Magnetism has two poles, called north and south.
* Two like poles exert repulsive forces on each other.
* Two opposite poles exert attractive forces on each other.
This is somewhat analogous to electric charges.
4.The poles of a bar magnet can be identified by using it as a compass (with other magnets it may be more difficult). 5.Materials that are attracted to a magnet are called magnetic materials. These materials are attracted to both poles.

Question 174

define a magnetic monopole

Answer 174

*Every magnet that has been observed has both a north pole and a south pole, thus forming a permanent magnetic dipole. *An isolated magnetic pole, such as a north pole in the absence of a south pole would be called a magnetic monopole. *Nobody has ever observed a magnetic monopole, although people have searched for them (some theories of subatomic particles say they should exist).

Question 175

Define the magnetic force

Answer 175

There exists a long-range magnetic force.
Define Magnetic Field 𝐵 as having the following properties:
1. A magnetic field is created at all points in space surrounding a current-carrying wire.
2.The magnetic field at each point is a vector. It has a magnitude, called magnetic field strength 𝐵, and a direction.
3.The magnetic field exerts forces on magnetic poles. The force on a north pole is parallel to 𝐵; the force on a south pole is opposite to 𝐵.

Question 176

Source of magnetic fields

Answer 176

The source of the magnetic field are moving charges. The magnetic field of a charged particle 𝑞 moving with constant velocity 𝑣 ≪ 𝑐 is given by the Biot-Savart law for point charge.
Biot-Savart Law is an equation for the magnetic field generated by a constant current

Question 177

derive the magnetic field of a current

Answer 177

Consider a short segment of wire. There is charge ΔQ in length of wire ΔS,

B(r)=µ/4π ((ΔQ)v×r)/r^2

(ΔQ)v=ΔQ ∆s/∆t=∆Q/∆t ∆s=I∆s

Therefore B(r)=µ/4π (I∆s×r)/r^2

Question 178

Amperes law

Answer 178

Amperes Law: ∮▒〖B ⃗∙ds ⃗= µI〗
One can show that this result:
*Is independent of the shape of the curve around the current.
*Is independent of where the current passes through the curve.
*Depends only on the total amount of current through the area enclosed by the integration path. using arguments similar to the ones use for Gauss’s Law.

Question 179

Lenz LAw

Answer 179

Lenz’s law (1834): There is an induced current in a closed, conducting loop if and only if the magnetic flux through the loop is changing. The direction of the induced current is such that the induced magnetic field opposes the change in the flux.
Faraday found that a current is induced when the magnetic flux through a conducting loop changes. The direction of the induced current is given by Lenz’s law.
A current needs an emf to provide the energy:
1. Motional emf is generated when a conductor moves.
2. A current is also created by changing the magnetic field with a stationary circuit. The emf associated with a changing magnetic flux, regardless of what causes the change is called induced emf.

Question 180

How is the magnetic flux through a conducting loop changed

Answer 180

The magnetic flux through a conducting loop can be changed in two fundamental ways:
1.The loop can move or expand or contract or rotate, creating a motional emf (due to magnetic forces!).
2.The magnetic field can change.

Question 181

How are EM waves generated

Answer 181

To make an EM wave, we need to generate an oscillating 𝐸 (or 𝐵) field. A dipole antenna does this by moving charges back and forth using an AC voltage source. The oscillating 𝐸 field generates an oscillating 𝐵 field, and so on. The result is an EM wave that spreads outwards from the antenna.

Question 182

Properties of EM waves

Answer 182

• 𝐸 and 𝐵 are perpendicular to each other.
• The propagation direction vem is along 𝐸 × 𝐵 (“Poynting vector”).
• The propagation speed is 𝑣em = 𝑐, the speed of light.
• The electric and magnetic fields oscillate in phase with each other, with magnitudes related by |E| = c|B| everywhere.
  These statements in fact apply to all electromagnetic waves, not just plane waves.