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EM Year 2 > Week 7 > Flashcards

Week 7 Flashcards

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1
Q

What is the continuity equation for charge?

A

Charge is conserved i.e. dp/dt = - Divergence of J

And holds at every point in space

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2
Q

How to obtain charge continuity equation?

A

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
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3
Q

What are the constraints on the distribution of charge and current?

A

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

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4
Q

How to derive charge continuity equation from Maxwell’s?

A
  1. Calculated divergence of Ampere-Maxwell’s law
    - divergence of curl of B = Divergence of ( (Muo J) + Epsilon oMu o dE/ DT))
  2. LHS = 0
  3. Substitute Gauss’ Law into RHS
    - Divergence of E = Rho/ Epsilon o
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5
Q

How many ways to EM fields and matter interact?

A

Only one - Lorentz force

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6
Q

How do you convert Lorentz Force Law from point charge to a volume?

A
  1. Substitute q for Rho d Tau
  2. Substitute Rho v for J
  3. Volume integral of (rho E Field + J cross B) d Tau
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7
Q

What is the Lorentz Force Law?

A

F = q( E + v x B)

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8
Q

What is the equation for EM force per unit volume on a charge distribution?

A

F = Rho E + J cross B

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9
Q

What is one formula for J?

A

Rho x V

  • Charge density x velocity at which charge are moving = flow of electric charge per unit area
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10
Q

How to calculate work using Lorentz force law?

A

DW = F . DL = F .v dt = q (E + v x B).v dt = qE.v dt

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11
Q

What is the rate of work done by EM fields on matter?

A

DW/ dt = Volume integral (E.J) d Tau

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12
Q

What is another term for the rate of work done by EM fields on matter?

A

Absorbed EM power

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13
Q

How to derive Poynting’s Theorem from Maxwell’s equations?

A
  1. Substitute Ampere-Maxwell law to get rid of J
  2. Apply Farady’s law
  3. Use vector identities
  4. Use divergence theorem
  5. Arrive at Poynting’s theorem
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14
Q

What’s another term for rate of work done by EM fields on matter derived using Maxwell’s equations?

A

Poynting’s theorem

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15
Q

What is Poynting’s theorem equation?

A

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

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16
Q

What is Poynting’s theorem in words?

A

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.

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17
Q

What is the Poynting Vector?

A

EM energy like current density

Direction of energy flux in an EM field

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18
Q

What is the continuity equation for EM energy in a vacuum integral form?

A

D/dt volume integral u d Tau = - Surface integral S. da

Where S = Poynting vector 1/ Mu o (E xB)

U = EM energy density

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19
Q

What is the continuity equation for EM energy in a vacuum differential form?

A

Du/dt = -Divergence of S

Where S = Poynting vector 1/ Mu o (E xB)

U = EM energy density

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20
Q

What is the formula for Poynting Vector?

A

S = Poynting vector 1/ Mu o (E xB)

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21
Q

What is u?

A

EM energy density

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22
Q

What is the formula for u?

A

EM energy density = 1/2 ( [E]^2 + 1/Mu o * [B]^2 )

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23
Q

What are the ways EM energy can be transferred/handled?

A
  1. Stored throughtout space - spatial density u
  2. Flow from one region to another - Poynting vector s
  3. Transferred to and from matter - Via Lorentz force
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24
Q

How does EM energy behave in a vacuum?

A

Without matter to transfer to or charges to “absorb” it is locally conserved

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25
What is total momentum?
EM + mechanical
26
What is DW/dt in a vacuum?
0 as there is no matter to transfer energy to.
27
What is radiation presssure?
Pressure from momentum exchange between photon and object
28
What’s 1D wave equation?
Second differential wrt z = 1/(V^2) * second differential wrt t
29
What are the solutions to the 1D wave equation?
F (z,t) = g(z-vt) + h(z+vt)
30
What do the solutions to the 1D wave equation represent?
1. 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
31
What is the sinusoidal wave equation?
F (z,t) = A cos (z - wt + delta) Where delta = phase shift
32
What is angular frequency formula?
W = 2PI/ T
33
What is frequency formula?
V = w/2PI
34
What’s the phase offset?
Displaces the wave along the z-vt axis - gives initial offset at t=0
35
What does k represent for waves?
Wave number - spatial angular frequency, indicating direction wave moves
36
What is euler’s identity?
E ^ (i theta) = cos theta + i sin theta
37
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)
38
What is the formula for complex phasor?
F tilde (z,t) - A tilde e ^ i(z-wt)
39
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
40
What is the formula for complex amplitude of the wave?
A tilde = a + ib = A e ^ i delta
41
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
42
Why is magnetic and electric Imelda wave equations identical?
Both are interdependent and related to c
43
What is speed of light formula relevant to em?
C = 1/ sqrt (mu o epsilon o)
44
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
45
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
46
What is E tilde sub o?
Complex amplitude vector - similar for B
47
What is the relevance of k.r in exponent of wave equation?
Dot product between 2 vectors = scalar
48
What is a plane wave?
A wave with uniform spatial dependence over every plane perpendicular to direction of travel
49
How is angular frequency related to wave speed?
K = +/- w/v
50
What does low k mean?
Long wavelength and short frequency
51
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
52
What does high k mean?
Short wavelength and high frequency
53
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
54
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
55
What are wave-fronts?
Points of constant phase
56
What is the general case electric field amplitude vector?
Eo tilde = Eo tilde (sup x) x hat + Eo tilde (sup y) y hat
57
What is the general case magnetic field amplitude vector?
Bo tilde = (1/c) (Eo tilde (sup x) y hat - Eo tilde (Supy) x hat
58
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)]
59
What is the relation of E, b and k vectors to each other?
Orthogonal
60
What is the wave-vector for a plane wave propagating along z-axis?
K = 0,0,k
61
What is the dot product for k and r for a z-propagating plane wave?
K.r = (0,0,k) . (X,y,z) = Kz
62
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
63
How to determine the phase of the wave?
Ratio of real to imaginary parts of wave
64
What is example of an x-polarised plane wave propagating along z in phasor notation?
1. Etilde (r,t) - Eo tilde x hat e ^ 1(kz-wt) 2. B tilde (r,t) = (Eo tilde/c) y hat e^ i (kz-wt)
65
What does is mean if a plane wave is linearly x-polarised ?
The E field varies in x plane, B varies in Y plane
66
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
67
What is the formula for electric field density?
(Epsilon o E^2) /2
68
What is the formula for magnetic field density?
B^2 / (2 Mu o)
69
How does circular polarisation happen?
When the E field vector (and sometimes B field) rotates in circular motion as wave propagates
70
In what direction is the Poynting vector?
Always in direction of wave propagation
71
Why is flux density in z direction?
Energy carried through space by wave
72
What are other equations for S?
S = c * epsilon o * Eo ^2 * cos^2(kz-wt+delta) z hat = cu z hat
73
What is another equation for u?
U = epsilon o * Eo ^2 * cos^2(kz-wt+delta) z hat = cu z hat
74
How to energy related quantised in EM fields vary?
Oscillate in time with a cosine-squared dependence
75
Why do we use the time averaged values?
Energy related quantities etc oscillate to fast as light so would encompass numerous cycles
76
What is the time-averaged value of a quantity over a period T?
= 1/T * Integral from 0 to T of f dt
77
What is the equation for the average value of a cosine squared over a complete cycle?
1/2
78
What is the equation for the average value of u over a complete cycle?
= 1/2 * epsilon o * Eo^2
79
What is the equation for the average value of a S over a complete cycle?
= 1/2 * c * epsilon o* Eo^2 z hat
80
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
81
What is the intensity of a plane wave equation?
I = [ ] = 1/2 * c * Eo^2 = 1/ (Eta o ) Eo^2
82
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
83
What is the formula for ETA o?
= E/B = SQRT( Muo/epsilon 0) = 1/ (c* epsilon o) = c * Mu o
84
What is the approximate value of ETA o?
~ 377ohms -> E field approx 377 x B field
85
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
86
What is the transversality condition?
The requirement that E and b fields in a plane wave are perpendicular to the direction of wave propagation
87
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
88
89
90
What is the relevance of the complex amplitude vector?
It tells us the polarisation of the wave, in which direction it’s oscillating
91
What are the units of the complex amplitude vector
V/m
92
94
What does w tell you?
the phase advance per unit time
95
How to find actual field?
Real part of wave
96
What is d/dx?
, ikx
97
What is d/dy
It’s iky
98
What does the wave equation tell us about k?
Kx^2 + ky^ +kZ^2 = (w/c) ^2 K = w/c
99
What is d/dt?
It’s -iw
100
What does the plane wave ansatz tell us about Maxwell’s equations?
K.E = 0 K.B = 0
101
What is B field in terms of E?
B = (k x E) / w
102
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
103
What does monochromatic mean?
104
What does time harmonic mean?
Single frequency
105
how to compute time integral?
= 1/T of integral of wave from 0 to T
106
What is Nabla equate to?
It’s iK , where k is a vector
107
Why do phasers only work for time harmonics?
Otherwise would have sin/cos which is not monochromatic unless use complex conjugates
108
What does k = k X hat mean?
Wave propagates along x
109
What does E tilde o = A y hat mean?
Wave is polarised along y with magnitude A
110
How to find real part of wave?
B (r,t) = Re { B (r,t) ] - use Euler’s identity and only Cos survives
111
How can you check RH rule if unsure?
Cross product
112
What is the formula for intensity?
Magnitude of energy flux (Poynting vector)
113
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EM Year 2 (10 decks)

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