Week 7 Flashcards
What is the continuity equation for charge?
Charge is conserved i.e. dp/dt = - Divergence of J
And holds at every point in space
How to obtain charge continuity equation?
DQ/ dt = - surface integral J. Da
- LHS -> Q = Triple integral of charge density (Rho d tau)
- RHS -> Divergence theorem to obtain Volume integral of (divergence of J) d Tau
What are the constraints on the distribution of charge and current?
The only constraint is the continuity equation whereas e.g. the divergence of current density increases, the rate of change in charge density decreases to maintain balance so charge is conserved
How to derive charge continuity equation from Maxwell’s?
- Calculated divergence of Ampere-Maxwell’s law
- divergence of curl of B = Divergence of ( (Muo J) + Epsilon oMu o dE/ DT)) - LHS = 0
- Substitute Gauss’ Law into RHS
- Divergence of E = Rho/ Epsilon o
How many ways to EM fields and matter interact?
Only one - Lorentz force
How do you convert Lorentz Force Law from point charge to a volume?
- Substitute q for Rho d Tau
- Substitute Rho v for J
- Volume integral of (rho E Field + J cross B) d Tau
What is the Lorentz Force Law?
F = q( E + v x B)
What is the equation for EM force per unit volume on a charge distribution?
F = Rho E + J cross B
What is one formula for J?
Rho x V
- Charge density x velocity at which charge are moving = flow of electric charge per unit area
How to calculate work using Lorentz force law?
DW = F . DL = F .v dt = q (E + v x B).v dt = qE.v dt
What is the rate of work done by EM fields on matter?
DW/ dt = Volume integral (E.J) d Tau
What is another term for the rate of work done by EM fields on matter?
Absorbed EM power
How to derive Poynting’s Theorem from Maxwell’s equations?
- Substitute Ampere-Maxwell law to get rid of J
- Apply Farady’s law
- Use vector identities
- Use divergence theorem
- Arrive at Poynting’s theorem
What’s another term for rate of work done by EM fields on matter derived using Maxwell’s equations?
Poynting’s theorem
What is Poynting’s theorem equation?
DW/dt = - d/dt volume integral (1/2 ( epsilon o [E]^2 + 1/Mu o [B]^2 )) d tau - surface integral (1/ Mu o (ExB).da
What is Poynting’s theorem in words?
The rate at which electromagnetic energy flows out of a given region in space is equal to the rate of work done on the charges within that region, plus the change in the energy stored in the electromagnetic field within the region; essentially, it describes how the energy within an electromagnetic field is transferred and transformed.
It’s conservation of energy either via outflow or conversion to other forms.
What is the Poynting Vector?
EM energy like current density
Direction of energy flux in an EM field
What is the continuity equation for EM energy in a vacuum integral form?
D/dt volume integral u d Tau = - Surface integral S. da
Where S = Poynting vector 1/ Mu o (E xB)
U = EM energy density
What is the continuity equation for EM energy in a vacuum differential form?
Du/dt = -Divergence of S
Where S = Poynting vector 1/ Mu o (E xB)
U = EM energy density
What is the formula for Poynting Vector?
S = Poynting vector 1/ Mu o (E xB)
What is u?
EM energy density
What is the formula for u?
EM energy density = 1/2 ( [E]^2 + 1/Mu o * [B]^2 )
What are the ways EM energy can be transferred/handled?
- Stored throughtout space - spatial density u
- Flow from one region to another - Poynting vector s
- Transferred to and from matter - Via Lorentz force
How does EM energy behave in a vacuum?
Without matter to transfer to or charges to “absorb” it is locally conserved
What is total momentum?
EM + mechanical
What is DW/dt in a vacuum?
0 as there is no matter to transfer energy to.
What is radiation presssure?
Pressure from momentum exchange between photon and object
What’s 1D wave equation?
Second differential wrt z = 1/(V^2) * second differential wrt t
What are the solutions to the 1D wave equation?
F (z,t) = g(z-vt) + h(z+vt)
What do the solutions to the 1D wave equation represent?
- Function depends on z-vt or z+vt
Where former propagates to right -> as t increases z increases to maintain constant;
and latter propagates to left
What is the sinusoidal wave equation?
F (z,t) = A cos (z - wt + delta)
Where delta = phase shift
What is angular frequency formula?
W = 2PI/ T
What is frequency formula?
V = w/2PI
What’s the phase offset?
Displaces the wave along the z-vt axis - gives initial offset at t=0
What does k represent for waves?
Wave number - spatial angular frequency, indicating direction wave moves
What is euler’s identity?
E ^ (i theta) = cos theta + i sin theta
What is the equation for a sinusoidal wave from euler’s?
Cos theta = Re {e ^(i theta) }
Re { A e ^ i (kz-wt + delta) } = Re { A e ^idelta * e^i (kz-wt)
What is the formula for complex phasor?
F tilde (z,t) - A tilde e ^ i(z-wt)
What is a complex phasor?
A complex number which represents a sinusoidal function whose amplitude, frequency and phase ae time invariant
I.e. screenshot in time
What is the formula for complex amplitude of the wave?
A tilde = a + ib = A e ^ i delta
What are the decoupled Maxwell wave equations?
By taking curl of both sides can separate electric from magnetic to get in a vacuum:
Laplacian of X = Mu o * Epsilon o * second derivative of X
- where X = E or B
Why is magnetic and electric Imelda wave equations identical?
Both are interdependent and related to c
What is speed of light formula relevant to em?
C = 1/ sqrt (mu o epsilon o)
What is a monochromatic plane wave?
A plane propagating in normal direction (e.g. moving in x direction with amplitude varying in y) with a single frequency
What is the formula for an EM monochromatic plane wave?
E tilde (r,t) = E tilde sub o = e ^ i (k.r-wt) and same for B
What is E tilde sub o?
Complex amplitude vector - similar for B
What is the relevance of k.r in exponent of wave equation?
Dot product between 2 vectors = scalar
What is a plane wave?
A wave with uniform spatial dependence over every plane perpendicular to direction of travel
How is angular frequency related to wave speed?
K = +/- w/v
What does low k mean?
Long wavelength and short frequency
When is a monochromatic plane wave a solution to EM wave equation?
In a vacuum if the ‘length’ of the k-vector is w/c
What does high k mean?
Short wavelength and high frequency
What do Maxwell’s equations tell us about the polarisation of a plane wave?
The first 2 tell us that the direction of oscillation of the electric and magnetic field vectors (given by Eo and Bo) is perpendicular to the propagation direction
I.E Light is a transverse wave
What do Maxwell’s equations tell us about the complex amplitudes?
The electric and magnetic complex amplitudes (Eo and Bo) are related to each other by a factor or c
Their directions are such that the vectors for E tilde, B tilde and k tilde form an orthogonal right handed triplet of vectors
What are wave-fronts?
Points of constant phase
What is the general case electric field amplitude vector?
Eo tilde = Eo tilde (sup x) x hat + Eo tilde (sup y) y hat
What is the general case magnetic field amplitude vector?
Bo tilde = (1/c) (Eo tilde (sup x) y hat - Eo tilde (Supy) x hat
When is non-linear polarisation revealed?
When wave is converted form a phasor notation into the “actual”field using E (r,t) = Re [ E tilde (r,t)]
What is the relation of E, b and k vectors to each other?
Orthogonal
What is the wave-vector for a plane wave propagating along z-axis?
K = 0,0,k
What is the dot product for k and r for a z-propagating plane wave?
K.r = (0,0,k) . (X,y,z) = Kz
What is the equation for EM plane wave propagating long the z axis?
X tilde (r,t) = Xo e ^ i(kz-wt)
Where X = E or B
How to determine the phase of the wave?
Ratio of real to imaginary parts of wave
What is example of an x-polarised plane wave propagating along z in phasor notation?
- Etilde (r,t) - Eo tilde x hat e ^ 1(kz-wt)
- B tilde (r,t) = (Eo tilde/c) y hat e^ i (kz-wt)
What does is mean if a plane wave is linearly x-polarised ?
The E field varies in x plane, B varies in Y plane
What is example of an x-polarised plane wave propagating along z in terms of actual fields?
E (r,t) = Eo cos (kz - wt + delta) x hat
B (r,t) = 1/c * Eo cos (kz - wt + delta) y hat
What is the formula for electric field density?
(Epsilon o E^2) /2
What is the formula for magnetic field density?
B^2 / (2 Mu o)
How does circular polarisation happen?
When the E field vector (and sometimes B field) rotates in circular motion as wave propagates
In what direction is the Poynting vector?
Always in direction of wave propagation
Why is flux density in z direction?
Energy carried through space by wave
What are other equations for S?
S = c * epsilon o * Eo ^2 * cos^2(kz-wt+delta) z hat = cu z hat
What is another equation for u?
U = epsilon o * Eo ^2 * cos^2(kz-wt+delta) z hat = cu z hat
How to energy related quantised in EM fields vary?
Oscillate in time with a cosine-squared dependence
Why do we use the time averaged values?
Energy related quantities etc oscillate to fast as light so would encompass numerous cycles
What is the time-averaged value of a quantity over a period T?
<f> = 1/T * Integral from 0 to T of f dt
</f>
What is the equation for the average value of a cosine squared over a complete cycle?
1/2
What is the equation for the average value of u over a complete cycle?
<u> = 1/2 * epsilon o * Eo^2</u>
What is the equation for the average value of a S over a complete cycle?
<S> = 1/2 * c * epsilon o* Eo^2 z hat
</S>
What is the intensity of a plane wave in words?
The time averaged power per unit area in W/m^2
I.e the norm of the time-averaged Poynting vector
What is the intensity of a plane wave equation?
I = [ <S> ] = 1/2 * c * Eo^2 = 1/ (Eta o ) Eo^2</S>
What does eta o represent?
The impedance of free space - a fundamental constant that describes how E and B fields ‘work together’ to transfer energy
- describes the relationship between the electric field and the magnetic field in an EM wave propagating through a vacuum
What is the formula for ETA o?
= E/B = SQRT( Muo/epsilon 0) = 1/ (c* epsilon o) = c * Mu o
What is the approximate value of ETA o?
~ 377ohms -> E field approx 377 x B field
What would an impedance higher than ETA signify?
E field component of EM wave is much higher compared to B field for same amount of energy
What is the transversality condition?
The requirement that E and b fields in a plane wave are perpendicular to the direction of wave propagation
Why are E and B fields perpendicular to motion?
As per Maxwell’s equations, field divergences in a vacuum = 0. If not perpendicular, E and B fields are moving so are sources/sinks
What is the relevance of the complex amplitude vector?
It tells us the polarisation of the wave, in which direction it’s oscillating
What are the units of the complex amplitude vector
V/m
What does w tell you?
the phase advance per unit time
How to find actual field?
Real part of wave
What is d/dx?
, ikx
What is d/dy
It’s iky
What does the wave equation tell us about k?
Kx^2 + ky^ +kZ^2 = (w/c) ^2
K = w/c
What is d/dt?
It’s -iw
What does the plane wave ansatz tell us about Maxwell’s equations?
K.E = 0
K.B = 0
What is B field in terms of E?
B = (k x E) / w
What is important step in calculating field/wave equations especially conversation from E to B?
K is a vector
- so need to cross this with r to ensure correct directions
What does monochromatic mean?
What does time harmonic mean?
Single frequency
how to compute time integral?
= 1/T of integral of wave from 0 to T
What is Nabla equate to?
It’s iK , where k is a vector
Why do phasers only work for time harmonics?
Otherwise would have sin/cos which is not monochromatic unless use complex conjugates
What does k = k X hat mean?
Wave propagates along x
What does E tilde o = A y hat mean?
Wave is polarised along y with magnitude A
How to find real part of wave?
B (r,t) = Re { B (r,t) ]
- use Euler’s identity and only Cos survives
How can you check RH rule if unsure?
Cross product
What is the formula for intensity?
Magnitude of energy flux (Poynting vector)
