Semester 2: Equations Flashcards
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
Define Stokes’ theorem
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.
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 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
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
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 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 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 is the equation for the energy stored in a capacitor in terms of the integrals for potential difference and charge?
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
What is Laplace’s equation in Cartesian coordinates?
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 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 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
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
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 polarisation in terms of dielectric susceptibility?
P = polarisation
ε₀ = permittivity of free space
χₑ = dielectric susceptibility
E = electric field
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 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 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
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
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
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 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
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
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
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 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
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
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
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 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 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
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
What is the equation for the magnetic dipole moment?
m = magnetic dipole moment
I = current
S = area vector
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
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
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
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
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
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 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
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
What is the equation for current in a closed circuit?
I = V/R
I = current
V = voltage
R = resistance
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
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
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
What is the equation for Faraday’s law?
E = electric field
Φ = ∫ B.dS = magnetic flux passing through an area, S
t = time
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 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
What is the equation for the change 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 is the Ampere-Maxwell equation?
∇ = del operator
H = magnetic field strength
J = current density
∂D/∂t = displacement current density
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 differential form of the 4 Maxwell relations?
What is the integral form of the 4 Maxwell relations?
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
How are the amplitudes of E and B fields related?
c = speed of wave
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
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
