Problems Flashcards
(197 cards)
1
Q
Key point to find potential of shell from Gaus
A
Set integral limits r to infinity
2
Q
2 key points to find charge from densities
A
- Integral is multiplied by charge density - rho, sigma etc
- Do not forget Jacobian
3
Q
How to find charge for solid e.g. sphere of radius R
A
- Charge density, ρ = total charge/total volume
= q/ 4 π/3 R^3
Or Q = ρ V
- Qenc = Q * (Volume inside Gaussian/volume inside sphere)
4
Q
What must you always remember when stating the field
A
Unit vector
5
Q
Energy formula for particles
A
- Total energy = K.E. A + K.E.B + Coulomb
6
Q
How to find simultaneous equations to solve for energy problems
A
- conservation of energy
- Conservation of momentum
7
Q
Radius used for dA in Gauss Law
A
Gaussian surface - usually r
8
Q
Radius used for charge
A
Small r inside shape
Big r outside shape
9
Q
Formula for energy stored in assembling spherical shell
A
W = 1/2 * εo ∫ σ dA
10
Q
Surface area of cone
A
πrS + πr^2
***S = slant height, not perpendicular height
11
Q
Steps to find potential difference of unusual shape
A
- Find potentials separately
- State potential equation; V = k ∫q/curly r
- Use diagram to define curly r, r and r’
- Find curly r in terms of unit vectors - consider change in coordinates
- Find magnitude of curly r
6.Define da in terms of coordinates - Use diagram to simply any terms
12
Q
Using curly R method to find potential or field at a distance from SOLID
A
- Take a slice and reduce to area integral
13
Q
How to find monopole moment
A
Sum of magnitude of charges
14
Q
How to find dipole moment
A
∑QiRi
* WITH UNIT VECTORS
15
Q
Potential at large r from pole moments
A
Potential = ∑pole = Vmonopole + Vdipole etc (kQ/r)
- Use Q = Qtotal for monopole and Q = p.r for dipole
- May need theta for dot product for latter e.g. 3qa Rhat.Z hat -> 3qa cos θ
16
Q
Force formula
A
QE
17
Q
Work formula
A
= qV
18
Q
Pattern of potential inside a conductor
A
Uniform
19
Q
Separation vector
A
From source to observer
20
Q
Calculate potential across changing E fields
A
= -kQ/ r
BUT different integral limits according to differing E fields
21
Q
Meaning a grounding wire making a shell’s potential = 0
A
Enclosed E field =0
22
Q
Capacitance from Electric field
A
- Gauss to find E field
- Inegrate to find potential
- C = Q/V
23
Q
What to remember for bound surface and volume charges
A
- Need conversion to spherical coordinates (or others if appropriate)
24
Q
Formula for bound volume charge
A
ρb = - ∇.P
**unit vectors and coordinate conversions
25
Formula for bound surface charge
σb = p.n =p cos \\\\\\\\\\\\\\\\\\\]’
**unit vectors and coordinate conversions
26
Volume bound charge if P is uniform
Zero
27
If put dielectric in eternal field Eext and it polarises (E induced), what is E total?
Eext + E induced
28
Electric field outside a volume with surface and volume charges (2 considerations)
1. Q = Volume x volume charges + volume x surface charges
2. Gaussian surface inside volume - sigma for charge
Gaussian surface outside volume - sigma and Rho for charge
29
How to substitute in unit vector e.g. for dipole
- r = l r l X R hat
30
Consideration asked to find electric field after volume/surface charge
- Sub in values for charges (remove sigma/rho)
31
Electric field
32
Gauss’ Law for Dielectrics
∇ . D = ρfree
D= displacement field
33
Which charges form displacement field?
Free charges
34
Which charges for external electric field?
Free charges
35
Define electric susceptibility
χe - how easily a material becomes polarized when placed in an electric field
36
Linear dielectric formula
P = ε 0χeE
37
Displacement field formula
D = ε 0E+ P
(where P = eoXoE)
38
Formula relating free charge to displament field
∫D. Da = Q free, enclosed
39
Features of bound charge (3)
- Arise from a material’s internal structure or dipoles
- Result from polarisation
- Can be measured and controlled
40
Best displacement field formula for known free surface charge Q
D = Q/A (n hat)
41
Best displacement field formula for known volume charge ρfree
∇ . D = ρfree
42
Best displacement field formula for known polarisation and electric field
D = ε E + P
43
Best displacement field formula for linear material
εE
44
Where are electric fields discontinuous?
- Boundary of surface of conductor and normal to surface - free charges
45
Charge density from electric field
∇.E = ρ / εo
46
Where are displacement fields discontinuous?
- At boundaries
47
Epsilon, ε , inside dielectric
ε = εo ( 1 + χ)
48
∇ (r hat/ r^2)
- 4 π δ^3 r
49
Electric field inside uniformly polarized sphere
E inside = - P/ (3 εo)
50
Relation of electric field to polarisation
Uniform and opposite to polarisation direction
51
diamagnetic
Create weak opposing magnetic field when exposed to external field - weak repulsion
52
Paramagnetic
Create weak aligned magnetic field in external magnetic field
53
Ferromagnetic
Permanent magnetisation
Create strong aligned magnetic fields, even without external magnetic field
54
Relate electrons to dia- para- and ferro- magnetic
- D: Paired electrons, small induced moment
- P: Unpaired electrons
- F: Unpaired electrons - collective coupling in unpaired spin
55
Relate magnetic susceptibility to dia- para- and ferro- magnetic
- D: χm <0
- P: χm > 0 but <<<1
- F: χm >>1 (large and positive)
56
Differential equation for current and current density
- dI = J x surface area dr
57
RHR for lorentz
Same as equation F = v x B
- Force
- Velocity
- B fields
58
Steps to find force on wire in magnetic field
1. Draw
2. F = I ∫ dl x B (cross product)
3. Write out with unit vectors
4. Find cross product of unit vectors
5. If given values for B or I in terms of coordinates sub in values e..g looking at line at x=-2 and B = kx, B= -2k
6. Integrate
7. UNIT VECTORS
59
Magnetic field due to long straight current carrying wire
1. Biot -Savart with curly r vertical to make RHR easier
60
Amperian loop for straight line
Circle
61
Amperian loop for slab or plane
Rectangle parallel and perpendicular to current ( sides will cancel)
62
Amperian loop for solenoid
Rectangle in plane parallel to solenoid axis with L/2 inside solenoid and L/2 outside (sides cancel)
63
Steps when usingAmpere’s
1. Draw with loop and all vectors (cross product)
2. Determine where fields cancel by symmetry
3. Draw 2D (axes with field and current)
4. Find current - integrate if non uniform
5.Write formula
5. Integrate
64
Heck for cross product unit vectors
Counter clockwise unit circle
65
Amperian loop for toroid
Circle inside and outside (hoops)
66
When to use magnetic field as dipole field
Distance to point of interest (curly r) >> loop dimensions
67
Magnetic dipole moment formula
M = I.A n hat
68
Bound charges vs bound currents
Charges - due to aligned electric dipoles (dielectrics)
Currents - magnetic material
69
Bound current formulas
- Surface K = M cross n hat (n = normal to surface, M= magnetisation vector)
- Volume Jb = ∇ cross M
70
Auxiliary field in words
Magnetic field intensity, relates magnetic field and magnetisation
71
Auxiliary field equation
H = (1/μo) B - M
72
Relate magnetic field magnetisation to applied magnetic field
Proportional M = χm H
χm = magnetic susceptibility
73
Formula for vector potential, A, from single dipole
A = (μo/4π) ( m cross curly r) / curly r^2
74
Formula for vector potential, A, for material (volume)
A = (μo/4π) ∫ ( m cross curly r) / curly r^2 D Tau
75
Important curly r formula
∇ 1/r = r hat/r^2
76
What is curl of magnetic field?
Current
77
Formula for total bound currents
- I bound ,volume = ∫ Jb dot dA
- I, bound, surface = ∫Kb dot dA BUT DO NOT INTEGRATE AS “dl” -> essentially just multiplication
78
When is bound volume current zero?
Uniform magnetisation
79
When is bound surface current zero?
Continuous B field at surface (no discontinuity)
80
Trick for curl in bound current
Cross product -> M x n = m sin θ
81
Bound surface currents for uniform sphere of rotating charge
- Kb = σ *velocity at each point = σ w r sin θ ( phi hat)
( where σ w r = M)
- Jb = 0
82
What does a cylinder with only surface current mimic?
Solenoid
83
What does a shell with only surface current mimic?
Loop
84
Area to use for da when integrating bound current to find I total
Crosssection it passes through
E.g. 2 Pi R in cylinder with Jb/Kb in Z/-Z hat direction
85
Magnetic field using Ampere’s Law and bound current
Volume current, ∇ x B = μo b
86
Magnetic field using Ampere’s Law and bound current
B ( r ) = μo/4π ∫ J(r’) dV’ x (r - r’) / l r - r’l^3
J(r’) = current density at source point
R -r ‘ = source to observation
87
88
Why is net force on dipole on uniform electric field zero?
Equal in magnitude
Opposite in direction
Equal distance so cancel
89
Why use auxiliary fields?
They separate the effects of free charges/currents from the material's response (polarization
P
⃗
P
, magnetization
M
⃗
M
).
90
91
Mutual inductance formula
Φ2 = M21 I1
92
Charge continuity formula and in words
= dρ/ dt = - ∇.J
Means whenever charge density changing wrt time, must be associated change in current flowing out of region
93
Derive integral form of charge continuity equation
1. dρ/ dt = - ∇.J
2. Divergence theorem
3. Sub in ∫ ∫ ∫ ρ τ = Q
94
Self inductance formula
Φ =LI
95
Define a phasor
Rotating vector used to represent oscillating field
96
Energy stored in a current in any loop of wire formula
W = 1/2 * L * I^2
97
Energy stored in magnetic field formula
W = (1/2 μo) ∫ ∫ ∫ l B l^2 d τ
98
How was Ampere’s Law inconsistent?
Inconsistent when electric fields changing
1. Charge continuity -> if take divergence of original Amperes -> curl of divergence=0 implying charge density can’t change
2. In charging capacitor can chose different
Amperian loops with different enclosed currents - Can have one in between plates -> B = 0, not true
99
How was Ampere Law problem solved
Addition of displacement current - non zero when take divergence
100
Maxwell’s Gauss/ from Coulomb’s in vacuum
∇. E = ρ/ εo
101
Maxwell’s - no name, biot savart
∇.B = 0
102
Maxwell’s Faradays’ from Biot Savart
∇ X E = -dB/dt
103
Ampere’s Correction Maxwell’s Law formula
∇ X B = μo J + μo εo dE/dt
104
Lorentz Force Law
F = q (E + v X B)
105
Maxwell’s Ampere’s with correction in macroscopic
∇ X H = J free + dD/dt
106
Which Maxwell’s laws are different macroscopically and why
- Ampere’s
- Gauss
Involve quantities affected by internal structure (polarisation and magnetisation)
107
Auxiliary fields
Mathematical fields used in EM to simplify Maxwell’s equations in materials
108
Maxwell’s Gauss macroscopic
∇. E = ρ free
109
Defiance constitutive relations of the material
Describe how a material responds to electric and magnetic fields
Links auxiliary fields (D,H) to primary fields (E,B)
110
Electric constitutive relation
D = εE
111
Magnetic constitutive relation
B = μH
112
Universal flux rule formula
E = - d Φ/dt
113
What is dA in magnetic fields
Vector area
- magnitude and orientation of surface perpendicular
114
What term does B field for loop moving in B field by rotating with angular speed w?
Cos wt
115
For rotating wire B field E.A
= E A cos θ
Where θ = angle between area and perpendicular component of area
116
What does “AC generators” imply?
Rotating with angular velocity
117
Steps to find EMF in rotation with angular velocity, w
1. Find Φ
Φ = B.dA = BA cos θ
2. Sub in θ = wt
Φ = BA cos (wt)
3. Faraday’s Law
EMF = - d/dt (Φ)
4. If needs be, can equate to EMF = IR to find current
118
Steps to find EMF in loop moving with speed v away from B field
1. Define area
A(t) = x . Y (t) = x. (L- vt)
L = original length of loop inside field
2. Define Φ
Φ = B.A(t) = B y ( L - vt)
3. Faraday’s Law
- EMF = - d/dt Φ = - d/dt ( By(L-vt) = Byv
119
What is conserved according to Maxwell’s equations
- Charge
- EM energy
- EM momentum
- Helicity…
120
Poynting theorem in words
The rate of change in EM energy inside a volume = - power absorbed/rate of work done on charges - flow rate of energy out of the volume
121
Which energy Poynting concerned with?
- EM to KE
122
What happens to rate of work done on charges in Poynting?
Converted to kinetic energy
123
Poynting energy formula
D/dt ∫∫∫ U dτ = - dW/dt - ∫∫ S. da
124
Which term in Poynting illustrates EM energy leaving volume?
∫∫ S. da
125
Which term in Poynting illustrates power?
dW/dt
126
Speed of light in universal constants
C = 1/ √ (μo εo)
127
What is k in wave notation (words and symbols)?
2π/λ
Wave number - Phase advance paper unit length
128
What is w in wave notation (words and symbols)?
Angular frequency - phase advance per unit time
129
Form of wave to use for phasor notation
F (z,t) = A e^iδ e^(ikx - it)
130
Define complex amplitude in symbols and words
A complex number that stores both the amplitude and the phase of a sinusoidal wave
à = Ae^iδ
131
Electric field wave equation from Maxwell’s Equations
Charged Fields Around maxwell’s Vehicle have Nothing Left
1. Take the curl of Faraday’s Law
2. Sub in Ampere’s law on RHS (for curl of B)
3. Use vector identity on LHS (for curl of E)
∇ x (∇x E) = ∇ (∇.E) -∇^2E
4. Use ∇ .E = 0 in a vacuum so only left with - ∇^2E on LHS
5. - ∇^2 E = μo εo (d^2E/dt^2)
6. Laplacian for vectors:
d2E/dx2 + d2E/dy2 + d2E/dz2 = 1/(1/ √ (μo εo))^2 d2E/dt2
132
What is Êo in EM plane waves?
Polarisation - constant
133
What is k vector dotted with itself?
W/c
134
Important to remember when finding E wave from plane wave solution
K.r in exponent
135
What is delta? (Ax e^iδx)
Phase oscillation of the x-component of the field
136
4 results from subbing E = Ēo e ^ (ik.r - iwt) into Maxwell’s equations?
The plane wave anstazs:
1. K.E = 0
2. K. B = 0
3. Kx^2 + ky^2 + kz^2 = w^2/c^2
4. B = (k X E) / w
137
Time harmonic
Constant frequency
138
Poynting Vector as cross product
S = E X B
139
What does Poynting vector describe?
- Direction and magnitude of EM energy in field
- Power/area carried by EM wave
140
Energy density formula (energeee densiteee)
1/2 EE and 1/2 MuBee
U (r,t) = 1/2 (Eo ll E(r,t) ll^2 + 1/μ ll B (r,t) ll^2 )
141
Orthognality in EM waves
B to E to k
142
Divergence theorem in words
Total flux of a vector field from a volume is equation to the integral of the divergence of the vector field over the volume
(Total water through a surface is determined by how much is entering or leaving)
143
Divergence theorem formula
∫∫∫ (∇.F) dV = ∫∫ F. dA
Volume integral = surface integral
144
Stokes Theorem formula
∫ F.dr = ∫∫ (∇ X F). dA
145
Stokes theorem in words
The surface integral of a closed loop is equal to the surface integral of the curl of a vector field
(The strength of rotation within a bound surface is related to how fast the water flows along the boundary)
146
Formula for electromagnetic force per unit volume on a charge distribution
F = ρE + J X B
147
Important concept for work by EM waves
B field does no work because perpendicular to velocity
148
What’s δ in plane wave?
Phase offset which displaces wave along z-vt axis
149
What does k vector tell you above the EM wave?
Direction of propagation
150
What does Eo tell you above the EM wave?
Strength of field and polarisation
151
What does w tell you above the EM wave?
- Propagation speed
- Can also derive energy
152
Transversality condition
E and B fields are always orthogonal to direction of propagation
153
Physical significance of transversality condition
- Energy flow in direction of wave travel
- Can only have transverse Em waves in free space (no longitudinal component)
154
Important plane wave derivative ansatzs to remember
1. d/dx = ikx = ∇
2. d/dt = -iw
3. ∇.F = ik.F
4. ∇ X F = ik X F
5. ∇^2 F = ∇.(∇F) = - (k.k)F
155
Which form of Maxwell’s equations to use for vacuum?
Microscopic
156
Which form of Maxwell’s equations to use for material?
Macroscopic as need auxiliary fields
157
How to write time averaged value
= limit as T -> infinity ∫U(t) dt
Half the real part of the wave’s product with one complex conjugated
- magnitude and unit vector left unchanged
- cos squared = 1/2
158
How to determine wave vector
Write as 3 column vector where k.r = [kx, ky, kz].[x,y,z] = [kx, 0,0]
159
How to find magnetic field from electric field phasor notation
B = 1/c * k hat * E field
RIGHT HAND RULE!!!!
160
How to check magnetic field unit vector from B field is correct?
S = 1/muo E X B
161
How to find actual part of field when complex amplitude is not real
Spilt Ao into amplitude and phase
Ão e^i Φ = Êo
E = Ão k hat e^i (kx - wt + Φ)
162
How to find actual part of field when complex amplitude is real
Take only real part of wave
Ēo y hat e^i(kx-wt) = Ēo y hat cos (kx - wt)
163
How to use energy density formula
Calculate square of fields separately- should get Eo^2 (cos)^
164
How does energy in E field compare to energy in B field?
Equal - can prove with c
165
Magnetic field for closed loop of wire
M = I A nhat
166
How to find area of loop for magnetic field if
1. Add areas vectorially - A total = A1 + A2 OR
2. ∫∫ dS
167
Creat expression for current in loop
- Use dq/dt where given line charge density = λ dl/ dt -> 2 options
- λ v -> λrw
- λ( R d θ) = λ R w
168
How to find magnetostatic field
- Biot Savart
- Ampere if symmetry especially if infinite
169
How to use Biot Savart for multiple ie. 2 rings d distance apart
Superposition
1. Bz of 1 ring at Z = 0 where Z = 0 is midpoint of them
B(z, w)
2. Superposition with shift of Z axis
Bz total = B(z-d,w) + B (z+d,w)
170
How to use Biot Savart for translations E.g. rotation
Superposition principle
- Find Bo at Z = 0
- Btotal = By + Bz etc
171
What to remember about s unit vector in cylindrical coordinates in circular loop of wire
S hat has angular dependence
172
Toolbox for Biot Savart
- Parameter of integration
- Source vector
- Observation vector
- curly R
- curly R unit vector
- path of integration E.g dl, dθ etc
- Current vector
173
When to apply symmetry in vectorial Biot Savart (always for EM 2)
AFTER I crossed with curly R
174
How to use superposition principle
1. Find solution for E.g. A= X at X =0
- take origin as most convenient point E.g. between 2 fields
2. Sub in values for all parts E.g. x = X-d/2, x = X+d/2
DO NOT SUBSTITUTE UNIT VECTORS, ONLY MAGNITUDES
175
Best way to calculate net dipole moments
As vector addition with unit vectors
176
Caution with vectors in formulas because….
Unit vectors are ONLY direction and disappear when dotted etc
Vectors have magnitude which remains
177
Which Maxwell’s equations are converted to integral form with Divergence theorem
- Gauss (E and B field)
178
Which Maxwell’s equations are converted to integral form with Stokes theorem
- Faraday’s and Ampere’s
179
What to remember when converting Faraday’s law to integral form?
At end take d/dt out of integral and convert ∫∫B.dA to Φ
180
What to remember when converting Ampere’s law to integral form?
- Jfree = Ifree enclosed when integrated over area
- Displacement field = Φ sub D when integrated over area
181
Which parts of Divergence theorem are vectors?
- The argument e.g. charge density, field
- dA
182
Which parts of Stokes theorem are vectors?
The argument and both “dS”s
183
How to determine how many dimensions to integrate over
Depends on shape given in question e.g in cylinder dV, loop of wire dl or dA depending on information
184
How to solve for Gaussian inside and outside volume
Inside: r of Gaussian = s of charge density
Outside: r of Gaussian = R of shape
185
How to convert sinusoidal wave to phasor
1. A (kx - wt + δ) = A e^iδ
-> Eo cos (kz - wt + π/2) Ŷ
2. e^iθ = cos θ + i sin θ
∴ e^(iπ/2) = cos (π/2) + i sin (π/2) = 0 + i and only looking at real part so = 0 gib m
-> Eo e^(iπ/2) Ŷ = i Eo Ŷ
186
Relevance of S=UC
- Energy conserved and flows in Poynting vector direction (flow = velocity x density)
- Same with charge, conserved and flow is current density
187
What to do before apply Coulomb’s law
Convert to line or surface integral
188
Steps when using Gauss Law to find field
1. State formula -> convert to 1D/2D formula if needed
2. State symmetry and Gaussian surface
3. Insert 2. into formula e.g 4pi R^2 for spherical
4. Form integral with dl’ as unit vector with column vector if found
3. Integrate -> just dl’ essentially
4. Solve for rR
189
How to check if potential is in Lorentz gauge
- Check if sum of time derivative of scalar potential and divergence of vector potential = 0
I.e. δV/δt + ∇. A = 0
190
Lorentz vs Coulomb gauge
Lorentz - both potentials, relativity
Coulomb - only vector potential
191
192
Shape of Gaussian according to symmetry
- Radially -> sphere
- Linearly (line) -> cylinder
- Sheet -> plane/pillbox
193
How to find curly r, r, r’
Use column vectors
194
How to derive dl
= differential length element of charge -> magnitude of parameterised dl
1. Identify path
2. Parameterise the path as needed e.g. circle => r (t) = (R cos t, R sin t, 0)
3. Differentiate position vector -> dr = -R sin t, R cos t, 0)
4. Find magnitude e.g. Sqrt [(-R sin t)^2 + (R cos t)^2 ]
5. Normalise vector
195
Find charge from line charge density, λ
dQ = λ dx
196
What to remember about charge when using Gauss
Leave in differential form * density for integral
197
Formula for parameterising dl
Ddl’ = ll dr’( t) / dt’ ll dt’
