Semester 2: Questions Flashcards
Give 2 examples of scalar quantities in electrostatics
- Electrostatic potential, V
- Potential energy, U = qV
Give 2 examples of vector quantities in electrostatics
- Electric field, E = - ∇ V
- Electric force, F = qE
What is the equation for the electric field?
E = electric field
∇ = del operator
V = electrostatic potential
What is the equation for the difference in potential for two points in space?
V = potential between two points
E = electric field
Electric field lines are always _______ to the electrostatic potential contours
Perpendicular
Define a conservative field
A vector field whose line integral only depends on the end points of the line, not the path taken to get there (the integral around all closed paths is zero).
Show that an electric field can be conservative using vector calculus
∇ = del operator
E = electric field
_ x _ = curl
∇ x E = -∇ x ∇ V = 0
Define Stokes’ theorem (electromagnetism)
The statements ∇ x E = 0 and ∫ E . dl = 0 are equivalent.
Define Gauss’ law
The total electric flux out of a closed surface is equal to the total enclosed electric charge divided by a constant.
What is the equation for Gauss’ law?
E = electric field
Q = charge
ε₀ = permittivity of free space
What is the equation for the electric field of a point charge?
E = electric field
q = charge
ε₀ = permittivity of free space
r = distance from point charge
What is the equation for the electric potential around a point charge?
V = electric potential
q = charge
ε₀ = permittivity of free space
r = distance from point charge
Define the divergence theorem
The flux of any vector field through a closed surface, S, can be related to the divergence of the vector field integrated over the volume enclosed by that surface.
What is the equation for the charge enclosed by a volume?
Q = charge
ρ = charge density
What is the definition of and the equation for the differential form of Gauss’ law?
The divergence of the electric field at every point in space is given by the charge density at that location, divided by a constant (for any volume).
∇ = del operator
E = electric field
ρ = charge density
ε₀ = permittivity of free space
Define the principle of superposition (for electric fields)
For a system of charged objects, the net electric field is equal to the vector sum of the fields of the individual charges.
What is the equation for the principle of superposition for the electric fields of a set of point charges?
E = electric field
q = charge
ε₀ = permittivity of free space
r = distance from point charge
Define the principle of superposition (for potentials)
For a system of charges, the total electrostatic potential is equal to the sum of the potential due to the individual charges.
What is the equation for the principle of superposition for the electric potentials of a set of point charges?
V = electric potential
q = charge
ε₀ = permittivity of free space
r = distance from point charge
What is a conductor?
Systems where charges are free to move. If an electric field is applied outside a conductor, the charges will move and redistribute inside the conductor until the force on them due to the electric field is zero.
What is the electric field inside a conductor in electrostatic equilibrium?
Zero
All points within a conductor are at the same ________ ________, also known as an ________________.
Electrostatic potential
Equipotential
What is the electric flux of a Gaussian surface enclosed within a conductor in electrostatic equilibrium?
Zero
Very close to the surface of a conductor, the electric field lines must be _________ to the surface of the conductor.
Perpendicular
What is the equation for the electric field close to the surface of a charged conductor?
σ = charge per unit area
ε₀ = permittivity of free space
n = unit vector perpendicular to the surface
What is the equation for capacitance?
C = capacitance
Q = charge
V = difference between electrostatic potentials
What does capacitance usually depend on?
The geometric arrangement of the conductors.
What is the equation for the electric field between the plates of a parallel-plate capacitor?
E = electric field
σ = charge per unit area
ε₀ = permittivity of free space
What is the equation for the electric potential between the plates of a parallel-plate capacitor?
V = electric potential (difference)
E = electric field
σ = charge per unit area
d = separation distance
ε₀ = permittivity of free space
What is the equation for the capacitance of a parallel-plate capacitor?
C = capacitance
Q = charge
V = potential difference
σ = charge per unit area
A = area of plates
d = separation distance
ε₀ = permittivity of free space
What is the equation for the work done to charge a capacitor from 0 to Q?
U = work done
Q = charge
C = capacitance
V = potential difference
What does the work done to charge a capacitor represent?
The energy stored in a capacitor charged to a given potential difference, V.
What is the equation for the energy stored in a capacitor in terms of the electric field present?
U = energy stored
ε₀ = permittivity of free space
E = electric field
What is the equation for the energy density of a uniform electric field?
u_E = Energy density
ε₀ = permittivity of free space
E = electric field
Define Poisson’s equation
∇ = del operator
V = potential difference
∇² = Laplacian operator
ρ = charge density
ε₀ = permittivity of free space
Define Poisson’s equation in a vacuum (free space)
∇² = Laplacian operator
V = potential difference
How can Poisson’s equation be derived?
By combining Gauss’ law with the equation for an electric field, E = - ∇ V.
What is Laplace’s equation in Cartesian coordinates?
Define Earnshaw’s theorem
It is not possible to find a local electrostatic potential minimum in all 3 spatial dimensions in free space as the following would have to be true (which isn’t possible if ∇² V = 0):
Any derivative of a solution to Laplace’s equation is ___ _ ________.
Also a solution
What is a boundary condition?
A location where the potentials or fields are ‘fixed’.
Define the uniqueness theorem
For a given set of boundary conditions, there is a unique solution to the electrostatic potential that satisfies Laplace’s equation.
What is the method of images?
A method of determining the potential due to a source in the presence of a hard boundary by replacing the hard boundary with an appropriately chosen source (such that the boundary conditions are still obeyed). These fictitious sources are known as images.
What is an electric dipole?
A system of two equal and opposite charges, ±q, separated by a small distance, s.
What is an electric dipole moment?
The product of the charges and separation distance between the charges in an electric dipole.
What direction does the electric dipole moment point in?
From negative to positive
What is the equation for the electric potential for a pair of point charges?
V = electric potential
q = charge
ε₀ = permittivity of free space
r = distance to charge
What type of symmetry does a dipole have?
Cylindrical symmetry
Do dipoles have a net charge?
No
What is the dependence of the electric field of a dipole on distance (from the centre of the dipole)?
1/r³
What are the equations for the electric field of a dipole in spherical polar coordinates (in r and θ directions)?
E = electric field
p = magnitude of the dipole moment
ε₀ = permittivity of free space
r = distance from the centre of the dipole
What is the dependence of the electric potential of a monopole on distance?
1/r
What is the dependence of the electric potential of a dipole on distance (from the centre of the dipole)?
1/r²
What is the dependence of the electric potential of a quadrupole on distance (from the centre of the quadrupole)?
1/r³
Define polarisability
The constant of proportionality between an electric field and an induced dipole moment, α.
What is a polar molecule?
A molecule with a permanent electric dipole moment due to the symmetry of its chemical bonding.
If a uniform, external electric field is applied parallel to an electric dipole, what is the net force acting on the dipole?
0
What is the equation for torque?
Γ = net torque
r = distance from pivot
F = force
What is the equation for the net torque on a dipole (due to an electric field)?
Γ = net torque
p = dipole moment
E = F/q = electric field
n = unit vector perpendicular to p and E
Why do dipoles have a net torque in an electric field?
Because the force acting on each charge is not always along the central axis of that dipole.
What is the equation for the electrostatic potential energy of a dipole?
U = potential energy
q = charge
V = electric potential of negative charge
V + ∆V = electric potential of positive charge
s = charge separation
E = electric field
(Second equation is valid if s is small)
What is the force on an dipole with potential energy, U?
F = force
∇ = del operator
U = potential energy
p = dipole moment
E = electric field
Define polarisation
The vector sum of all the dipole moments per unit volume, P.
P = polarisation
p = dipole moment
Does polarisation occur when there is no electric field? Why?
No because the molecules are aligned randomly and have nothing to align with.
Does net polarisation occur when there is an electric field? Why?
Yes because the molecules tend to orient along the E direction due to torque.
What is a dielectric?
An (electrically) insulating material that can be polarised in an electric field.
What is a non-polar dielectric?
Materials that become induced electric dipoles in an applied field.
What is the equation for polarisation?
P = polarisation
N = number of aligned dipole moments
p = qx = dipole moment
V = volume
What is the equation for the magnitude of polarisation?
P = polarisation
N = number of aligned dipole moments
p = qx = dipole moment
A = area
x = length
What is the equation for the electric field inside a dielectric?
E = electric field
σ = charge per unit area
P = polarisation
ε₀ = permittivity of free space
What is the equation for the potential difference between two capacitor plates (in terms of the polarisation of the dielectric between the plates)?
V = potential difference
E = electric field
d = distance between plates
σ = charge per unit area
P = polarisation
ε₀ = permittivity of free space
What is the equation for polarisation in terms of dielectric susceptibility?
P = polarisation
ε₀ = permittivity of free space
χₑ = dielectric susceptibility
E = electric field
The dielectric susceptibility is always _ 0.
≥
What is the equation for the relative permittivity (the dielectric constant) of a material in terms of the dielectric susceptibility?
εᵣ = relative permittivity
χₑ = dielectric susceptibility
What is the equation for the charge per unit area for a homogeneous dielectric in a uniform electric field of a parallel-plate capacitor?
σ = charge per unit area
ε₀ = permittivity of free space
εᵣ = relative permittivity
E = electric field
What is the equation for the capacitance of a parallel-plate capacitor when the plates are separated by a dielectric?
C = capacitance
Q = charge
V = potential difference
σ = charge per unit area
A = area
E = electric field
d = separation distance
ε₀ = permittivity of free space
εᵣ = relative permittivity
What is the equation for the ratio between a capacitor separated by a dielectric and one separated by a vacuum?
C = capacitance
εᵣ = relative permittivity
What does relative permittivity depend on?
Frequency
What is the equation for surface charge per unit area for an arbitrary shape?
σ = charge per unit area
P = polarisation
n = unit vector perpendicular to the surface
What is the equation for the sum of all polarisation charges across a surface and within a volume?
σ = charge per unit area
ρ = charge per unit volume
What is the equation for the total electrostatic potential produced by polarisation charges in an arbitrarily shaped dielectric?
P = polarisation
n = unit vector perpendicular to the surface
ε₀ = permittivity of free space
r = distance from dielectric
What does the charge density within a medium consist of?
- Free conduction charges
- Bound, induced polarisation charges
What is the differential form of Gauss’ law for both conduction charges and polarisation charges?
∇ = del operator
E = electric field
ρ = charge density
P = polarisation
ε₀ = permittivity of free space
What is the equation for the charge density of conduction charges?
ρ = charge density
∇ = del operator
ε₀ = permittivity of free space
E = electric field
P = polarisation
What is the equation for the electric displacement field?
D = electric displacement field
ε₀ = permittivity of free space
E = electric field
P = polarisation
What is the equation for the electric displacement field in terms of the relative permittivity?
ε₀ = permittivity of free space
εᵣ = relative permittivity
E = electric field
Compare an electric field, a polarisation field, and a displacement field in a parallel-plate capacitor containing a uniform dielectric
What is the general expression for the energy density associated with a material in an electric field (in terms of the electric displacement field)?
u = energy density
ε = ε₀εᵣ = permittivity
E = electric field
D = electric displacement field
P = polarisation
How can the change in the displacement field be calculated when travelling between two dielectric media perpendicularly?
1) Model a cylindrical Gaussian surface enclosing the boundary between the dielectrics.
2) The flux is the difference between the two media multiplied by the surface area of the cylinder.
3) Using Gauss’ law, this is equal to the charge per unit area multiplied by the surface area.
4) Hence D₂ - D₁ = σ.
How can the change in the displacement field be calculated when travelling parallel two dielectric media?
1) Use the electrostatic circuital law for a closed path crossing the boundary between the two media.
2) Add the integral of each side of the path.
3) The sum of these paths is equal to 0, hence, ∆D = 0 and D₂ = D₁.
How does the displacement field change when travelling between two dielectric media at an oblique angle?
Permittivity affects the perpendicular component but not the parallel component, hence, the angle that the displacement field leaves at is different to the angle it enters at.
What is the equation for the force on a charge moving through a magnetic field?
F = force
q = charge
v = velocity
B = magnetic field
What is the equation for current density?
J = current density
ρ = charge density
v = mean velocity of charges
Current is the ____ of the current density through an area, S.
Flux
What is the equation for current (in terms of current density)?
I = current
J = current density
S = area
What is the magnitude of the current in a wire if the cross-sectional area it travels through is uniform?
I = NevS
I = current
N = number of conduction electrons
e = electron charge
v = average speed
S = cross-sectional area
What is the equivalent equation to Gauss’ law in magnetostatics?
B = magnetic field
S = area
What is the magnetic equivalent of electric charge?
There is no magnetic equivalent of electric charge.
If a magnetic flux coming out of a box is measured, there must be ________ magnetic flux entering the box.
Equal
What is the equation for the divergence of a magnetic field?
∇ = del operator
B = magnetic field
What is the equation for Amperes circuital law for magnetostatics?
B = magnetic field
µ₀ = permeability of free space
I = current passing through the area enclosed by the path
c = closed path
What is the right-hand rule for magnetostatics?
The direction of the magnetic field, B, due to a long straight wire carrying a current, I, can be found by curling the right hand into a thumbs-up shape. The thumb represents the direction of current and the curled fingers represent the direction of the field.
How can the magnetic field of a long straight wire carrying a current, I, be found?
The integral around a circular path of radius, r, can be found surrounding the wire in cylindrical coordinates. The magnitude is constant in all directions around the circle so does not have to be included in the integral.
What is the equation for the magnetic field of a long straight wire of radius, a, when r > a?
B = magnetic field
r = radius of closed path
µ₀ = permeability of free space
I = current passing through the area enclosed by the path
What is the equation for the magnetic field of a long straight wire of radius, a, when r < a?
B = magnetic field
r = radius of closed path
µ₀ = permeability of free space
I = current passing through the area enclosed by the path
a = radius of wire
Why is the equation for a magnetic field of a long straight wire different inside vs outside the wire?
Inside the wire, only the current that passes through the area πr² contributes to the magnetic field so a fraction of the current has to be calculated (chosen area/ wire area).
Outside the wire, all of the current contributes to the magnetic field.
How can the magnetic field of a solenoid be calculated using Ampere’s law?
A square path that encloses an amount of current (equal to current * number of loops) should be chosen and integrated over using Ampere’s law because this is the location where the magnetic field is non-zero. All parts of this loop are 0 except for the section inside the loop. This has a constant magnitude so the integral for Ampere’s law is equal to magnetic field * length of path inside the solenoid.
What is the equation for the magnetic field of a solenoid using Ampere’s law?
B = magnetic field
µ₀ = permeability of free space
N = number of coils
I = current passing through the area enclosed by the path
L = length of path inside the solenoid
n = number of coils per unit area
When is Ampere’s law valid for a solenoid?
Far from the ends of the solenoid.
Define Stoke’s theorem
The integral of a vector field around a closed path is equal to the curl of that vector field integrated over the surface enclosed by the path.
What is the equation for Stoke’s theorem for a magnetic field?
B = magnetic field
∇ = del operator
c = closed path
S = area
What is the equation when Stoke’s theorem is combined with Ampere’s law?
∇ = del operator
B = magnetic field
µ₀ = permeability of free space
J = current density
S = area
What is the equation for the differential form of Ampere’s law?
∇ = del operator
B = magnetic field
µ₀ = permeability of free space
J = current density
What does the differential form of Ampere’s law show?
It shows that the variation in the magnetic field at every point in space is determined only by the current density at that point.
What is the equation for the Biot-Savart law (the current element formula)?
dB = change in magnetic field
µ₀ = permeability of free space
I = current
dl = infinitesimal length
r = shortest distance from the current element to a point, P, where the field is measured
____ _______ _____ can be thought of as the ‘magnetism’ equivalent of the electric field from a point charge. Both follow an inverse square law
The Biot-Savart law
What can the Biot-Savart law be used for?
It can be used with the principle of superposition to calculate the distribution of electric currents.
What is the equation for the force on a charge, q, moving in an electromagnetic field?
F = force
q = charge
E = electric field
v = velocity
B = magnetic field
What is the equation for the work done by an electromagnetic field to accelerate a charge over a small distance?
W = work done
F = force
dl = small distance
q = charge
E = electric field
v = velocity
B = magnetic field
In the equation for the work done by an electromagnetic field to accelerate a charge over a small distance, the electrostatic component is the work done by the electric force and is equal to _____ _____ _____. The magneto static component does ___ ______ on the charge.
Kinetic energy gained
No work
What is the equation for the work done on a charge by a magnetic force?
q = charge
v = velocity
B = magnetic field
dt = change in time
What is the equation that describes the motion of a free particle in a magnetic field?
m = particle mass
v = velocity
r = radius ( = mv/qB)
q = charge
B = magnetic field
How does a free particle move in a magnetic field?
It undergoes circular orbital motion.
What is the equation for the cyclotron frequency (the frequency of the orbit of a particle in a magnetic field)?
ω = orbital frequency ( = v/r)
q = charge
B = magnetic field
m = mass
What component of a particles velocity is unaffected by a magnetic field?
A component parallel to the magnetic field. The path of a particle would be helical in this case.
What is the effect of applying an electric field to a conductor?
Current will flow
What is the effect of applying a magnetic field perpendicular to the current in a conductor called?
The Hall effect
What is the Hall effect?
The application of the Lorentz force. It is where a current and magnetic field are applied to a conductor perpendicular to one another. The moving charges will feel a v x B force that acts perpendicular to both v and B. This causes the charges to be directed towards one side of the conductor and a potential difference is built up until the electric and magnetic forces balance.
What is the equation that relates current to the speed of charged particles in a conductor?
I = current
n = number of charged particles per unit volume
q = charge
v = velocity
A = cross-sectional area of the conductor
What is the equation for the Hall voltage?
V = Hall voltage (= vBw)
I = current
B = magnetic field
n = number of charged particles per unit volume
e = charge
d = thickness of conductor (=A/w)
A = area
w = width
What is an application of the Hall effect?
Hall sensors: used to sense and measure magnetic fields by utilising the proportionality of the Hall voltage to the perpendicular magnetic field.
What is the equation for the force on a section of current-carrying wire in a magnetic field?
dF = force on section
q = charge
v = velocity
B = magnetic field
I = current (= nAqv)
What is the equation for the force acting on a uniform wire of length, L, in a magnetic field?
IL x B
I = current
L = length
B = magnetic field
For a closed electric circuit in a uniform magnetic field, what is the net magnetic field experienced?
Zero
What is the equation for the torque on a current-carrying wire loop in a uniform magnetic field?
Γ = torque
r = radius of loop
F = magnetic force
L = length
I = current
B = magnetic field
S = area
What is the equation for the rotational potential energy of a current-carrying wire loop in a uniform magnetic field?
U = rotational potential energy
Γ = torque
θ = angle between B-field and area vector (perpendicular to surface)
I = current
S = area
B = magnetic field
When is the rotational potential energy of a current-carrying wire loop in a uniform magnetic field at its lowest?
When the magnetic field is parallel to the area vector (that is perpendicular to the surface). Here, torque is equal to 0.
When is the torque of a current-carrying wire loop in a uniform magnetic field at its highest?
When the magnetic field is in the plane of the loop (perpendicular to S).
When is the rotational potential energy of a current-carrying wire loop in a uniform magnetic field at its highest?
When the magnetic field is antiparallel to the area vector (that is perpendicular to the surface). Here, torque is equal to 0.
What is the equation for the magnetic dipole moment?
m = magnetic dipole moment
I = current
S = area vector
What is the equation for the translational force that occurs when a current-carrying wire loop is in a non-uniform magnetic field?
F = force
U = energy
m = magnetic dipole moment
B = magnetic field
How is the torque of an electric dipole different to the torque of a magnetic dipole?
LHS = electric dipole
RHS = magnetic dipole
How is the potential energy of an electric dipole different to the potential energy of a magnetic dipole?
LHS = electric dipole
RHS = magnetic dipole
How is the translational force on an electric dipole different to the translational force on a magnetic dipole?
LHS = electric dipole
RHS = magnetic dipole
How is the field on an electric dipole different to the field on a magnetic dipole?
LHS = electric dipole
RHS = magnetic dipole
What is the equation that relates the magnetic field to a vector potential?
B = magnetic field
∇ = del operator
A = vector potential
Why is the relation between the magnetic field and a vector potential defined as B = ∇ x A?
Because this equation automatically satisfies Gauss’ law for magnetostatics.
What is the equation for the current of a single electron orbiting a nucleus (following a circular path)?
I = current
e = electron charge
T = time period
v = velocity
a = radius
What is the magnetic dipole moment of a single electron orbiting a nucleus?
m = magnetic dipole moment
I = current
a = radius
e = electron charge
v = velocity
L = mₑva = angular momentum
mₑ = electron mass
What is the magnetic dipole moment of a single electron orbiting a nucleus in terms of quantised energy levels?
L = nħ
n = energy level = 0, ±1, ±2, …
m = magnetic dipole moment
mₑ = electron mass
µ_B = Bohr magneton
How much does the magnetic dipole moment of an orbiting electron contribute to its total atomic magnetic moment?
It contributes very little as the quantum mechanical property, spin, contributes the most to the angular momentum.
What is the spin magnetic dipole moment of an electron?
±µ_B
Define magnetisation
The vector sum of magnetic dipole moments per unit volume.
What is the equation for magnetisation?
M = magnetisation
mᵢ = magnetic dipole moment
What is the magnetic dipole moment for a piece of magnet with a cross-sectional area, S?
m = magnetic dipole moment
I_B = bound current
S = cross-sectional area
What is the equation that relates magnetic dipole moment to magnetisation?
m = magnetic dipole moment
M = magnetisation
dl = length of magnet piece
S = cross-sectional area
What is the equation for bound current?
I_B = bound current
M = magnetisation
dl = length of magnet piece
What is the equation for Ampere’s law in magnetostatics when the closed path crosses the centre of a current loop and a magnet?
B = magnetic field
dl = length
µ₀ = permeability of free space
I = current
I_B = bound current
How can Ampere’s law for magnetostatics for a current loop and a magnet be modified to include magnetisation?
B= magnetic field
µ₀ = permeability of free space
I = current
M = magnetisation
Define magnetic field strength
A vector field that provides a way of writing Ampere’s law such that the effect of magnetic materials is automatically included
What is the equation for magnetic field strength?
H = magnetic field strength
B = magnetic field
µ₀ = permeability of free space
M = magnetisation
What is the equation for Ampere’s law in terms of magnetic field strength?
H = magnetic field strength
I = current
What is the equation for Stoke’s theorem in terms of the magnetic field strength?
H = magnetic field strength
∇ = del operator
What is the equation for the differential form of magnetic field strength?
∇ = del operator
H = magnetic field strength
J = conduction current density
What is the equation for the differential form of magnetisation?
∇ = del operator
M = magnetisation
J_B = bound current density
Define magnetic susceptibility
A measure of how readily a material develops a magnetisation when a magnetic field is applied.
What is the equation for magnetic susceptibility?
M = magnetisation
χₘ = magnetic susceptibility
H = magnetic field strength
What is the equation for the relative permeability of a material?
µᵣ = relative permeability
χₘ = magnetic susceptibility
What is the equation for permeability in terms of the magnetic field and magnetic field strength?
B = magnetic field
µ = permeability
H = magnetic field strength
Magnetic susceptibility can be either _________ or ___________.
Positive
Negative
What is a paramagnetic material?
A material whose magnetic susceptibility > 0. This means that the magnetic dipole moment aligns parallel with the magnetic field strength.
What is a diamagnetic material?
A material whose magnetic susceptibility < 0. This means that the magnetic dipole moment aligns antiparallel with the magnetic field strength.
In most materials, χₘ __ 1. This means that µᵣ ~ 1.
«
What is a ferromagnet?
A regular magnet, either a hard paramagnet or a soft electromagnet, that has a very large magnetic susceptibility (> 1000).
What is a superconductor?
A material with a magnetic susceptibility of -1 up to a critical value of magnetic field. This means that B = µ₀(H + M) = 0 inside the superconductor and that energy is conducted with no resistance.
Using the differential form of Gauss’ law, what is the equation that relates magnetic field strength to magnetisation?
∇ = del operator
H = magnetic field strength
M = magnetisation
What type of magnetic fields is Gauss’ law true for?
B-fields but not M-fields or H-fields.
Whenever there is a source of H, there is a sink of _ and vice-versa. These sources and sinks correspond to _____ __ _ _________.
M
Poles of a magnet
What pole does the H-field emanate from?
The north pole (and runs to south pole)
What pole does the M-field emanate from?
The south pole (and runs to north pole)
What units are the B-field measured in?
Tesla
What units are the H-field and M-field measured in?
A/m
What are the units of permeability of free space?
Henrys per metre = Vs/Am
What are the equations for electric and magnetic fields in a steady (time-independent) state?
∇ = del operator
E = electric field
H = magnetic field strength
J = current density
I = current
Magnetic fields generate ________ _______ and vice versa.
Electric fields
What is the equation for the potential difference of a conducting rod in a uniform magnetic field when the electric forces balance the magnetic forces?
V = potential difference
e = charge
v = velocity
B = magnetic field
L = rod length
Electric force: eE = eV/L
Magnetic force: F = evB
There is no current flow in a moving conductor when ___________ force and ________ force are balanced in a magnetic field.
Magnetic
Electric
What is the equation for current in a closed circuit?
I = V/R
I = current
V = voltage
R = resistance
Define electromotive force (EMF)
The electric potential induced in a stationary charge in a changing magnetic field or a conductor moving in a uniform magnetic field.
What is the equation for the EMF due to motion in a uniform magnetic field?
EMF = electromotive force
E = electric field
v = velcocity
B = magnetic field
L = length of moving rod
Give an example of when an EMF will be generated for a moving conductor in a magnetic field?
A rod rolling along a fixed conducting track in the field.
What is the equation for the rate of change in area of a conducting track as a rod rolls along it?
A = area
t = time
v = velocity
L = length
What is the equation for the rate of change of magnetic flux in a square coil?
Φ = BA = magnetic flux
B = magnetic field
A = area
t = time
v = velocity
L = length of leading/trailing side
The rate of change of flux for a square coil entering a region of uniform magnetic field is ___________.
Positive
The rate of change of flux for a square coil inside a region of uniform magnetic field is ___________.
Zero (so current falls to 0)
The rate of change of flux for a square coil leaving a region of uniform magnetic field is ___________.
Negative
Define Faraday’s law
Any change in magnetic flux through a coil, for any reason, will induce an electromotive force.
What are 3 examples things that cause a change in magnetic flux?
- Magnet moving through a circuit loop
- Coil moving around a bar magnet
- Switching a solenoid on/off next to a circuit loop
What is the equation for Faraday’s law?
E = electric field
Φ = ∫ B.dS = magnetic flux passing through an area, S
t = time
Define Lenz’s law
Any current flow through a closed circuit is in a direction such that it opposes the magnetic flux change causing the EMF.
What is the differential form of Faraday’s law?
∇ = del operator
E = electric field
B = magnetic field
t = time
What is the equation that relates magnetic flux to current?
Φ = magnetic flux
L = self-inductance
I = current
What is the result of a change of current in a magnetic field?
A change in magnetic flux which causes an induced voltage.
Define induced voltage
A voltage that causes the circuit to tend to resist any change in the electrical current passing through it (the resulting current opposed the change of magnetic flux).
This is an example of Lenz’s law.
What is the equation for induced voltage?
V = induced voltage
L = self-inductance
dI/dt = rate of change of current
What is the equation for the electric field in a time-varying magnetic field?
E = electric field
∇ = del operator
V = scalar electrostatic potential
A = magnetic vector potential
t = time
What is the equation for the total flux of current out of a surface when the electric field is changing with time?
J = current density
S = area
Q = charge
t = time
In what situation would the electric field be changing with time?
If charge was leaking out of a surface
What is the equation for the charge enclosed in a surface?
Q = charge
ρ = charge density
V = volume
What is the charge continuity equation?
∇ = del operator
J = current density
ρ = charge density
t = time
What does the charge continuity show?
It shows that the current diverges away from locations where the charge density is locally decreasing (i.e. total charge is conserved).
What does the charge continuity equation show about Ampere’s law?
It shows that Ampere’s law is only valid if the charge density is not changing with time.
What is the Ampere-Maxwell equation?
∇ = del operator
H = magnetic field strength
J = current density
∂D/∂t = displacement current density
What does the Ampere-Maxwell equation show?
It is a form of Ampere’s law when the charge varies with time.
What is the equation for displacement current density?
∂D/∂t = displacement current density
ε₀ = permittivity of free space
∂E/∂t = rate of change of electric field
∂P/∂t = rate of change of polarisation
What is the integral form of the Ampere-Maxwell equation?
H = magnetic field strength
J = current density
∂D/∂t = displacement current density
I = current due to conduction charges
I_D = flux of displacement current density
What is the integral form of the 4 Maxwell relations?
What is the differential form of the 4 Maxwell relations?
How can conversions be made between the integral and differential forms of the 4 Maxwell relations?
Using the divergence theorem (for surface integrals and divergences) and Stoke’s theorem (for path integrals and curls).
How do the Maxwell equations change in a vacuum?
ε₀ = µ₀ = 1
ρ = J = 0
What is the wave equation for an electric field in a vacuum?
∇ ² = Laplacian operator
E = electric field
c = speed of light in a vacuum
What is the wave equation for a magnetic field in a vacuum?
∇ ² = Laplacian operator
B = magnetic field
c = speed of light in a vacuum
What is the plane wave solution to the wave equation for an electromagnetic field?
E = electric field
E₀ = amplitude
ω = angular frequency
t = time
k = wavenumber
z = displacement
What happens to the plane wave solution to the wave equation for an electromagnetic field when Gauss’ law is applied?
It becomes 0
What type of wave is an electromagnetic wave?
A transverse wave
The E and B fields associated with an EM wave are __________ to one another.
Perpendicular
How are the amplitudes of E and B fields related?
c = speed of wave
How do the wave equations for electric and magnetic fields change in a dielectric media?
The speed of the wave is no longer the speed of light in a vacuum.
What is the equation for the velocity of a wave in a dielectric medium?
v = velocity
ε = permittivity
µ = permeability
What is the equation for the refractive index of a dielectric?
n = refractive index
Define Poynting’s theory
A vector expression of conservation of energy associated with an electromagnetic field. The Poynting vector points in the direction of propagation of the wave.
What is the equation for the Poynting vector?
N = Poynting vector
E = electric field
H = magnetic field strength
What is the equation for the divergence of the Poynting vector?
∇ = del operator
N = Poynting vector
u_EM = electromagnetic energy density
E = electric field
J = current density
What is the equation for electromagnetic energy density?
u_EM = electromagnetic energy density
B = magnetic field
E = electric field
ε = permittivity
µ = permeability
What are the equations for the E and B fields of a plane wave polarised in the x-direction?
What is the equation for the total electromagnetic energy density associated with a plane wave?
u_EM = electromagnetic energy density
E = electric field
ε = permittivity
What is the equation for the time-averaged Poynting vector associated with a plane wave?
N = Poynting vector
v = velocity
u_EM = electromagnetic energy density
