Newton to Einstein Flashcards
Define the unit Vector
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.
Define the Dot Product
C = a . b = |a||b|cos(x) , where x is the angle between a and b when they are ‘tail to tail’
Define the cross product
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.
Define the term Kinematics
Kinematics is the study of the relationship between an object’s position, speed and acceleration.
Define average velocity
Average velocity is the ratio between the displacement S and the time interval Δt in which the displacement occurs. V = S/Δt
Define instantaneous velocity
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
Define constant velocity
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.
Define acceleration
Acceleration is the rate of change of velocity.
Express the position of an object as a position vector in terms of its constituent vectors parallel to x and y.
r = xi +yj
Express the velocity of an object as a vector in terms of its constituent vectors parallel to x and y.
v ̅=ⅆx/ⅆt i ̂+ⅆy/ⅆt j ̂ = vxi + vyj
Express the instantaneous acceleration of an object as a vector in terms of its constituent vectors parallel to x and y.
a = ⅆv/dt = (ⅆv_x)/ⅆt i ̂+(ⅆv_y)/ⅆt j ̂
What is the defining characteristic of projectile motion, excluding air resistance.
Acceleration parallel to y only.
Define angular displacement
Final Θ - Initial Θ
Define angular velocity ω
ω = dΘ/dt
Rate of change of angular displacement
Define angular acceleration (alpha)
Rate of change of angular velocity
What are the respective signs of angular velocity for acw motion and cw motion
ω > 0 for acw motion , ω < 0 for cw motion
What are the defining characteristics of uniform circular motion
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.
Define velocity for uniform circular motion
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
What Force is responsible for the circular motion of an object.
Centripetal Force given by:
F = mv^2 / r
What is the characteristic feature of centripetal acceleration
It is always perpendicular to the velocity of an object exhibiting circular motion
Define Centripetal Accelerstion
|a|= ω^2 r = v^2 / r
Define the principle of superposition of forces
The net force on an object is the vector sum of the forces acting on it.
Give the relationship between angular acceleration and tangential acceleration
alpha = a_t / r
Give the rotational kinematic equations
ω_f = ω_i+ αΔt
θ_f = θ_i Δ_t + 1/2 α(Δt)^2
(ω_f)^2 = (ω_i)^2 + 2αΔθ
Define Gravitational Force
Gravity: Long range force proportional to the mass. Acts on all objects (moving or at rest) in a gravitational field. F = -GMm/r^2
Define spring force
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.
Define the Tension force
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.
Define the Normal Force
Reaction force that always acts perpendicular to the plane.
Define Static Friction
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.
Define Kinetic Friction
Kinetic Friction: Acts opposite to the direction of motion. Fk = μ_k . n
Define Drag Force
Acts opposite to the direction of motion of an object in a fluid. D=1/2 CρAv^2 ( (-v ) /|v| )
Define Newtons First Law
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)
Define Newtons Second Law
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.
Define Newtons third law
If object A exerts a force on object B, object B also exerts an equal and opposite force on object A. (action-reaction pair)
Define the frictional Force
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.
Define Apparent weight
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)
Define Terminal velocity
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.
Key Characteristics for Tension in strings and pulleys.
-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.
What Force is responsible for a car’s turning circle when travelling round a banked track
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.
Define conservation of Energy
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.
Define Mechanical energy
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.
Define Conservative Forces
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.
General Strategy for Equilibrium problems
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
Define Work done by a conservative force
W = -ΔU
Where ΔU is the change in potential energy.
Give the relationship between conservative forces and potential energy.
W = -ΔU and W = int[F ds]
Therefore -dU/ds = F
General Strategy for Dynamics problems
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
General Strategy for Problems concerning objects in contact
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.
Derivation of F=ma
F = dp/dt
dp/dt = d/dt (mv) = m.dv/dt +v.dm/dt = m.dv/dt = ma
Derivation of impulse
let J = int[F dt] = int[ma dt] = int[m. dv/dt dt] = mv2-mv1 = Δp
impulse-momentum theorem
Define Conservation of momentum
Total momentum of an isolated system is conserved
Define Impulse
Impulse if the change in momentum of an object. It is the area under a Force-Time graph.
Define Work Done
work done by a force is the dot product of force and displacement
W = F. Δs = FΔscos(x)
Define Power
Power is the rate of change of doing work. P = dw/dt = F . V
Define a rigid body
Rigid bodies do not change shape in motion these include; translation, rotation and translation and rotation.
About what point will a rigid body rotate when not constrained to a fixed axis
its centre of mass
How do you calculate centre of mass
x_cm = 1/M int[x dm]
where dm = M.dx / L
Define the kinetic energy of a rigid body undergoing rotation
K_rot = 1/2 I ω^2
Define moment of inertia
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
Give the expression for the kinetic energy of a body exhibiting translational and rotational motion
K = 1/2mv^2 + 1/2 I ω^2
Sum of translational kinetic and rotational energy
How do you find the moment of inertia of an extended object.
Sum over all the moments of inertia of constituent points of the extended object
Define rolling without slipping
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.
What is the perpendicular axis theorem
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
What is the parallel axis theorem
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
What Three variables does the ability of a force to cause rotational motion depend on
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
Define Torque
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.
Define net torque on an object
τ_net = I α
where I is moment of inertia and α is angular acceleration.
It is the rate of change of angular momentum.
Define Static Equilibrium
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
What rule determines the direction of the angular velocity vector ω
Right hand rule. Thumb indicates direction of ω, Fingers indicate direction of rotation.
Define Angular momentum of a point particle about the origin
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.
Define conservation of momentum
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.
What is the angular momentum for a particle exhibiting circular motion
L=mvr, this is because v and r are always tangential.
Explain the procession of a gyroscope
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
Define angular momentum for an extended object rotating about one of its axis
Angular momentum is given by:
L = I ω.
Define Oscillatory Motion
Repetitive motion about an equilibrium position. It can be periodic or non periodic
What is the defining characteristic of simple harmonic motion
For an oscillation to exhibit SHM the acceleration of the system must be opposite and proportional to the displacement.
Give the equations of displacement, velocity and acceleration for mass on a spring subject to x = A, at t = 0 with V = 0
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
Give the disaplacement equation for a mass on a spring that doesnt start from x=A at t=0
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
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
w^2 = k/m and T = 2π√(m/k)
Give the form of the solution and the physical definitions for the solutions of differential equations corresponding to damped oscillators
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
Define a travelling wave
An organised disturbance that propagates through space with a well-defined wave speed. Waves transfer energy but do not transfer matter.
Define a mechanical wave
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.
Give the defining characteristics of transverse and longitudinal waves respectively
Transverse waves oscillate perpendicular to the propagation and longitudinal waves oscillate parallel to the propagation.
Give the wavespeed equation
v=fλ
the wave speed is also described by
v= ω/k
Give the wavespeed for waves on a string
v= √(T/µ) where T and µ is the tension in the string and the linear density of the string respectively.
What do snapshot graphs of a wave show.
Oscillatory displacement against linear displacement for a given time
What do history graphs of a wave show
Oscillatory displacement against time for a given point in space
What type of source gives a sinusoidal wave
Wave sources that oscillate with simple harmonic motion generate sinusoidal waves. The frequency of the wave is the frequency of the oscillating source.
Define the wavenumber
wave number k= 2π/λ
define the angular frequency
angular frequency ω= 2π/T
Give D(x,t) for a sinusoidal wave travelling in the positive and negative directions respectively.
D(x,t) = Asin(kx-ωt+φ) +ve
D(x,t) = Asin(kx+ωt+φ) -ve
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?
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)
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?
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)
Describe fixed end reflection
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.
Describe free end reflection
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.
Describe what happens when a pulse crosses a boundary from a less to more dense string.
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.
Give the principle of superposition
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.
Define interference
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.
What leads to a standing wave
The superposition of two sinusoidal waves travelling in opposite directions, but with the same frequency, wavelength and amplitude leads to a standing wave.
Describe a standing wave
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.
Give the equation for a standing wave
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.
Give the boundary conditions and their results for a standing wave on a string.
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
What are possible standing waves called that meet the boundary conditions
Possible standing waves that meet the boundary conditions are called normal modes of the string.
How are sound waves created from a loudspeaker
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.
How does the speed of sound depend on the properties of the medium
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.
Describe the helium effect on the voice
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).
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)
D = D1+D2=asin(kx_1-ωt+φ_10 )+asin(kx_2-ωt+φ_20 )=asinφ_1+asinφ_2
Using sinα±sinβ=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]
Give the conditions for maximum constructive interference for two identical sources.
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
Give the conditions for maximum destructive interference for two identical sources.
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.
Define the doppler effect
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 .
Give the frequency detected by a moving observer for a stationary source
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
Give the frequency detected by a stationary observer for a moving source
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
Define the reference frame
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
Define an inertial reference frame
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.
Define an event
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.
What is Einstein’s basic postulate of relativity and its implications
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.
Define the relativity of simultaneity
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.
What is a light clock.
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.
Define proper time
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.
Define the proper length
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 .
Define the spacetime interval
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.
Define relativistic momentum
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.
give the energy-momentum invariant expression
E^2-(pc)^2=(mc^2 )^2=〖E_0〗^2
Continuous spectra
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)
Discrete spectra
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.
Oil drop experiment
- “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.
Give the results of Phillipp lenards experiment
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.
Give the classical interpretation of lennards results
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
What is debroglies proposition and equation
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.
What is the general result for a particle of mass m confined in a region of length L (particle in a box)
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
Give Bohr’s model of quantisation
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.
Give the classical equation for an electron in orbit about the nucleus
(mv^2)/r=e^2/(4πε_0 r^2 )
Derive the angular momentum of an electrons orbit
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.
Derive the bohr radius
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.
Deduce the born rule using a double slit experiment of monochromatic light
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.
What is the normalisation condition of the wavefunction
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).
Define a wave packet
in quantum mechanics, a wave packet is a localised oscillation analogous to a classical particle.
Explain the Heisenberg Uncertainty Principle
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.
Compare Classical and Quantum stationary states
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.
Give the conditions on physical solutions of the shrodinger equation.
- 𝜓 (𝑥) must be a continuous function – no jumps.
- The derivative of 𝜓 (𝑥) must be continuous except where U(x) is infinite
- 𝜓 (𝑥) = 0 where U(X) is infinite (forbidden regions)
- ∫(-∞)^∞▒〖P(x)dx=∫(-∞)^∞▒〖|ψ(x)|^2 dx〗〗=1
- 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.
Define the classically forbidden region
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.
Describe the particle in a box model
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
Describe the finite potential well
Defined by zero potential in some given interval with high but finite potential elsewhere.
What does the penetration describe about the wave function
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
What is the quantum correspondence principle
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.
What are the General principles to sketch wave functions
- 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).
What is the law of conservation of charge
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.
Properties of Insulators
*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
Properties of conductors
*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.
Describe electric dipoles
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.
Describe coulombs law
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.
A charge q at a point in space where the electric field is 𝐸 experiences an electric force give by…
F=qE
Define the electric field model
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.
Give the properties of electric field lines
*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.
Define linear charge density
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).
Define symmetry in the context of the E-Field of an extended object
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.
Define electric flux
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.
Give the properties of conductors in electrostatic equilibrium
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.
Define the electric field
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|
Define the electric potential
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.
How do you find the potential from the electric field
- Draw a diagram and identify the point at which you wish to find the potential. Call this position i.
- Chose the zero-point of the potential, often at infinity. Call this position f.
- Establish a coordinate axis from i to f along which you already know or can easily determine the electric field component Es.
- Integrate the following equation to find the potential: -int[E.ds]
Define an equipotential
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.
Explain why charge density is higher at points with a high degree of curvature for extended objects.
- 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.
Give the potential difference and capacitance across a capacitor
〖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
Energy stored in capacitor
U_c=1/2 C(ΔV)^2=1/2 (ε_0 A)/d(Ed)^2=1/2 εAdE^2
Energy density of capacitor
U_E = U_c / volume = U_c/ Ad
Describe what happens to the charges in a conductor when a conductor in inside a non zero E field
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.
Describe electric current density
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
Describe the relationship between electric current density and the electric field weakly out of equilibrium
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
Define kirchoffs loops law
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.
Define kirchoffs junction law
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.
Drude Model
- 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.
Problems with the drude model
- 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
Give the 5 basic experimental magnetic results
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.
define a magnetic monopole
*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).
Define the magnetic force
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.
Source of magnetic fields
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
derive the magnetic field of a current
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
Amperes law
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.
Lenz LAw
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.
How is the magnetic flux through a conducting loop changed
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.
How are EM waves generated
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.
Properties of EM waves
- 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.
