Electricity and Magnetism
(Fall 2008)
Text: Introduction to Electrodynamics, by David J. Griffiths
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Day # 1: Coulomb's Law and Electric Field
Covers: pp. 58–61, 8–10 (Section(s): 2.1.1–2.1.3, 1.1.4)
Goals:
1. Be able to use Coulomb's Law for point charges to calculate the force on a charge.
2. Know when Coulomb's Law is valid
3. Understand and be able to calculate electric field E
4. Introduce and understand r, r', 𝓇, q, Q notation.
Homework # 1:
1. Answer the following short questions in a sentence or two.
a. What is meant by the terms ‘test charge’ (Q) and ‘source charges’ (q)?
b. What is the principle of superposition?
c. What is electrostatics?
d. What is Coulomb's Law?
e. When is Coulomb's Law valid?
f. What is the definition of Electric Field?
g. How does 𝓇 relate to r and r′? (Show equation and draw a diagram.)
2. Problem 2.1 on page 60
Answer: a. 0N, b. kqQ ⁄ r² pointing toward missing charge, c. 0N, d. same as b
Hint: Parts a and c use symmetry while b and d build on these with the superposition of a negative
charge (= –q) to cancel out one of the charges. The easiest way to prove the results for a and c is to
exploit the symmetry of the problem using a proof by contradiction. Assume there exists a net force
in an arbitrary direction (with in the plane of the charges in this case). What happens to the
direction of that force when you rotate it? Is this possible considering symmetry?
3. Four point charges q = 20 µC each, are on the corners of a square of length 4 m. Find the force on a
point charge Q = 100 µC located 3 m just above the center of the square.
Answer: 3.08N
Hint: The direct approach is to use 1.28 on page 10. Alternatively you can use a symmetry argument
similar to the previous problem and/or use the results of the next problem.
4. Problem 2.2 on page 61
Answer: a. E = 2kqz ⁄ (z² + ¼ d²)3 ⁄ 2 in the ẑ-direction, b.
Hint: Neither answer will be zero, nor will they have same magnitude nor direction. In order to
determing the large distance limit assume that d = 0 where ever you are adding it to z. The same is
also true for d² with z² of course.
Read: pp. 61–64 (Section(s): 2.1.4)
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Day # 2: Coulomb's Law for Continuous Charge Distributions
Covers: pp. 61–64 (Section(s): 2.1.4)
Goals:
1. Understand charge density and its relationship to charge.
2. Be able to use the integral form of Coulomb's Law for simple cases.
Homework # 2:
1. Answer the following short questions in a sentence or two.
a. What is the relationship between the total charge Q of a given volume and ρ? Q of a surface and σ?
Q of a line and λ?
b. What is the equation for Coulomb's Law in its integral form (for a continous 3d charge
distribution)?
c. In Coulomb's equation given above which of E, ρ, 𝓇 are functions of r? of r′?
d. How is it that E is only a function of r when the rest of the terms either depend on r′ or both r and
r′
2. A thin triangular plate (with corners located at (0,0,0), (2m,0,0), and (0m,2m,0)) is located in the x-y
plane. a. What is the total charge on the plate if the plate has a uniform charge density σ = 0.003 C ⁄ m²?
b. What is the total charge on the plate if σ = Dxy, where D = 0.003 C ⁄ m⁴
Answer: a. 0.006 C; b. 0.002 C
Hint: You don't need to evaluate the integral for part a. Be careful with the surface integral in the
second part. In particular since the area is not a square the first integral you perform will have an
upperlimit that is a function of the second variable. See your rigid body textbook under center of
mass (and moment of inertia) to see how this integral is performed. One straightforward method is
to break the area integral into strips with the first integral giving you the ‘length’ of each strip and
the second integral adding up these ‘lengths’ to determine the final ‘area’.
3. A sheet of charge lies in the z=0 plane and occupies the area 0<x<2m and 0<y<2m. The surface
charge density is σ = 2x(x² + y² + 4)3 ⁄ 2 C ⁄ m⁶. Determine the electric field 2 m above the sheet on the
z-axis. For full credit make sure to write down the values of 𝓇 , 𝓇=||𝓇𝓇 ||, and 𝓇 ⁄ 𝓇³ and the equation you are
using for Coulomb's law.
Answer: E=(−32 ⁄ 3 k,−8 k, 16 k)
Hint: Remember Coulomb's law is a vector equation. You must do three separate integrals (one for
each component of E involving the corresponding component of 𝓇 ). If you don't get three easy
integrals for Coulomb's Law then you have done something wrong (probably in your determination
of 𝓇). Don't get the idea that all Coulomb's problems are as easy as this. Real world problems are
almost always very difficult.
4. Problem 2.6 (p. 64)
Answer: 2πσkz[1 ⁄ z − 1 ⁄ √R² + z²] ẑ
Hint: This problem can be found in many general physics text books. Simply copying the solution will
not help you, though, should this problem appear in a test. In order to obtain full credit you need to
explicitly write down what r, r' and 𝓇 are. Symmetry can be used to simplify this problem
dramatically (the symmetry is similar to problems 3 and 4 of the previous homework). Integrate in
the φ direction first. (It should be trivial.) The integral over r is only a little harder (use substitution).
Read: pp. 65–67, 26–27 (Section(s): 2.2.1(start), 1.3.1(surface integrals only))
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Day # 3: Electric Field Lines, Electric Flux, and Outflow
Covers: pp. 65–67, 26–27 (Section(s): 2.2.1(start), 1.3.1(surface integrals only))
Goals:
1. Be able to sketch and interpret Electric Field Lines diagram from source charges.
2. Given an electric field line diagram be able to determine the relative strength of the Electric field at a
given location as well as the location of all charges.
3. Understand the concept of an area vector (in particular for the infinitessimal diferential areas).
4. Understand the physical meaning of Electric Flux through a surface area and be able to calculate it.
5. Understand the physical meaning of outflow of Electric field for a closed surface and be able to
calculate them.
6. Understand and be able to show that the outflow of the surface of a composite volume is equal to the
sum of the outflows through the surfaces of its component parts.
Homework # 3:
1. Answer the following short questions in a sentence or two.
a. Explain how to determine the Electric field at a given point and the location and relative
magnitudes of all charges given an Electric Field diagram.
b. Highlight and label the sign of all regions in the following Electric field diagram where there is
charge. (There are no electric field lines in the shaded region.)
Hint: there are 2 positive and 2 negative regions of charge.
c. What is meant by an area vector? (What is its magnitude and direction?)
d. What is the equation for the Electric flux, Φ, through an area A given that the surface is flat and
that E is constant (magnitude and direction) over that surface? What are the units of Electric flux?
e. What is the equation for the Electric flux through an arbitrarily shaped surface for arbitrary electric
field E?
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2. Answer the following questions using the following diagram. Here the electric field is constant over all
space E=10kN ⁄ C x̂. The scale of the drawing is such that the distance between the electric field lines is
10 cm as is the height (out of the paper) of each area (represented by the thick grey lines).
a. Estimate the electric flux Φ through each area ‘by eye’.
Answer: 0 kNm² ⁄ C (horizontal area only), 0.2 kNm² ⁄ C (rest)
Hint: Use the area that is perpendicular to the Electric field lines as a reference.
b. What is the area vector for each area? (Write down numerical values for the x and y components of
the area vector for each area.) Draw the vector for each area on the diagram.
c. Calculate the electric flux Φ for each area mathematically.
Hint: If your solution is longer than one line each you are doing it the hard way.
d. Do your answers agree?
3. Problem 1.29 on page 28.
Answer: The flux through the bottom square = −12
The flux does not depend only on the boundary line
The total flux through the cube = 32
Hint: The question about the dependence on the boundary line is confusing. Example 1.7 and
Problem 1.29 are two different surfaces with the same boundary (the circumference of the bottom
face of the cube). For the flux to be ‘only dependent’ on the boundary the flux must be the same
for all surfaces having that boundary. You can't prove that the flux is ‘only dependent’ on the
boundary with just two surfaces but you can disprove it if you can find two surfaces with the same
boundary that have different flux through them.
4. ‘Prove’ Guass' Law (for one static point charge q) that the outflow (flux outward) through any closed
surface (Φ=∯E·da) will be q ⁄ ε₀ for any surface that encloses q and 0 for any closed surface that does
not enclose q in the following manner.
a. First show that Φ=∯E·da = q ⁄ ε₀ for the simple case of the of a spherical surface of radius a
centered around the point charge q.
Hint: If this isn't trivial then you are doing it the hard way (or incorrectly).
b. Show that the outflow of the surface of a composite volume is equal to the sum of the outflows
through the surfaces of its component parts.
Hint: Prove this for the case of 2 connected arbitrarily shaped volumes using the definition of flux
and properties of integrals. You can infer all other non-pathological cases from this special case.
This proof is straight forward (needing only a few lines) and relies on the fact that the flux
through the common suface cancels. It is important that you explain and understand why they
cancel.
c. Show that the outflow due to a charge q centered at r=0 through a closed surface having the 6 sides
represented by the equations (r=r₀, r=r₀+∆r, θ= θ₀, θ= θ₀+∆θ, φ= φ₀, φ= φ₀+∆φ) equals zero for
arbitrary r₀, ∆r, θ₀, ∆θ, φ₀, ∆φ.
Hint: The most difficult part of this problem is understanding, visualizing and drawing the surfaces
involved. The math is trivial since all surfaces are either parallel or perpendicular to the Electric
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field (which is in the r̂ direction) Like the previous part of this problem understanding the
solution requires understanding the conventional direction of the area vector for outflow.
d. Show that the outflow through the surface of any non-pathological volume that does not include q
must be zero.
Hint: Use the fact that any non-pathological volume may be made from a large combination of
small volumes of the previous part of this problem.
e. Show that the outflow through any closed surface that encloses q must be q ⁄ ε₀
Read: pp. 68, 70–74 (Section(s): 2.2.1(mid), 2.2.3)
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Day # 4: Gauss' Law and its application
Covers: pp. 68, 70–74 (Section(s): 2.2.1(mid), 2.2.3)
Goals:
1. Know the integral form of Gauss' Law
2. Know when Gauss' Law is valid
3. Be able to use Gauss' Law in cases with high symmetry
Homework # 4:
1. Answer the following short questions in a sentence or two.
a. What is the integral form of Gauss' Law?
b. When is Gauss' Law valid. How does this compare to Coloumb's Law
c. What 3 types of symmetries is Gauss' Law (in the integral form) most useful for?
2. Problem 2.10 (p. 69)
Answer: q ⁄ 24ε₀
Hint: This problem can be solved in one line uses Guass' Law and geometry.
3. Problem 2.12 (p. 75)
Answer: ρr ⁄ 3ε₀ r̂
4. Problem 2.16 (p. 75)
Answer: i. E = ρs ⁄ 2ε₀ ŝ
ii. E = ρa² ⁄ 2ε₀s ŝ
iii. E = 0
Read: pp. 28–29, 31–33, 17–18, 68–70 (Section(s): 1.32, 1.34, 1.24, 2.2.1(end), 2.2.2)
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Day # 5: Charges as Outflow sources, Outflow Density, Divergence, Fundamental
Theorem of Divergence, and the Differential form of Gauss' Law
Covers: pp. 28–29, 31–33, 17–18, 68–70 (Section(s): 1.32, 1.34, 1.24, 2.2.1(end), 2.2.2)
Goals:
1. To recast Gauss' Law in a differential form
2. Know that charges are sources of Electric field (outflow)
3. Understand that Gauss' Law states that the amount of outflow through a closed surface is proportional
to the charge in the volume.
4. Therefore the outflow density ≡ Outflow ⁄ Volume (∯E·da ⁄ τ) is proportional to charge ⁄ volume = ρ
5. Know that the divergence of a vector field ∇ ·v is defined as the outflow density for an infinitessimal
volume of that vector field. This is known as the Fundamental Theorem of Divergence.
6. Know, remember, and be able to use the Cartesian representation of the divergence.
7. Be able to calculate the divergence of vector fields in Cartesian, spherical, and cylindrical coordinates.
8. Understand Gauss' Law can be recast into a differential form using the Fundamental Theorem of
Divergence.
9. Be able to calculate the charge distribution that produces a given Electric field.
Homework # 5:
1. Answer the following short questions in a sentence or two.
a. What is the fundamental Law of Divergence?
b. What is the Cartesian representation of the divergence of E, ∇ ·E?
c. Is the equation for the representation of the divergence the same for all coordinate systems?
d. Highlight all regions in the following diagram that have a non zero divergence. (There is no electric
field in the shaded region.) Indicate whether the outflow is positive or negative for each region.
e. What is the equation for the differential form of Guass' Law?
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2. Problem 1.15 (p. 18)
Answer: a. 0 b. y+2z+3x c.?
Hint: Since the answers are given for parts a) and b) I need you to show enough steps to show me that
you know how they were calculated.
3. Problem 1.16 (p. 18)
Answer: 0 everywhere except at r = 0.
Hint: Remember that each coordinate system has it own representation of the divergence. Don't
forget the sketch and to explain what the ‘discrepency’ with the sketch is and how to resolve it. (As
a hint to why notice that you are dividing by r² in the equation for the divergence.)
4. Problem 1.32 (p. 33)
Answer: 48 = 48
5. If the vector fields of Problem 1.15 (p. 18) were equations for the electric field E then what are the
charge distributions ρ that produced these Electric fields? Does it matter whether or not these electric
fields are electrostatic?
Read: pp. 90–91 24–25, 77–82 (Section(s): 2.4.1, 1.3.1(start), 2.3.1–2.3.2)
Ignore the discussion about the curl (∇
∇ ×) for now. What is important here is that E is conservative (or that
∮E·dl=0 for all closed paths. We will be discussing the gradient of V ∇ V shortly. For now it is enough to
know that the gradient is defined as the inverse of the line integral such that if V=−∫E·dl then E=−∇
∇ V.
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Day # 6: Conservative Vector Fields and the Potiential (V)
Covers: pp. 90–91 24–25, 77–82 (Section(s): 2.4.1, 1.3.1(start), 2.3.1–2.3.2)
Goals:
1. Understand and remember the relationship between the electrostatic potential V and electric potential
energy
2. Know (and remember) the definition of the potential (V) in terms of the electric field E
3. Know what is meant by a conservative vector field.
4. Be able to calculate line integrals
5. Know (and remember) the definition of the circulation of a vector field ( E) for a given closed path.
(Answer: ∮E·dl)
6. Be able to demonstrate that a particular Electric field is not conservative by calculating the circulation
for a given loop.
7. Know that the electric field E of any electrostatic charge distribution is conservative. But, that the E of
moving charges does not have to be conservative
Homework # 6:
1. Answer the following short questions in a sentence or two.
a. What is the definition of the electrostatic potential in terms of the Electric field?
b. What is the relationship between the electrostatic potential (V) and the electric potential energy
(U)?
c. What units is electric potential measured in?
d. Answer the following True/False Questions:
i. The electric field must be conservative in order to have a well defined potential.
ii. The electric field is conservative for all electrostatic cases.
iii. The electric field is conservative for all electrodynamic cases.
iv. The electric field is not conservative for all electrodynamic cases.
e. What mathematical requirement is needed for a vector field to be conservative?
f. What is the definition of the circulation of a vector field E for a given loop C?
g. When measuring voltage why must you always use 2 leads on your voltmeter?
h. How much energy does a 12V battery use to push 2 Coulombs of charge through a pair of
headlights?
2. Problem 1.28 (p. 28) plus e. Can you determine whether or not v is conservative from d. alone.
Answer: a. 4/3 b. 4/3 c. 4/3 d. 0 e. the answer to d. is indicative but not a proof, explain why.
Hint: Remember that the direction you integrate along the path is important. Integrating along the
same path in the opposite direction will give the same value but with the opposite sign.
3. Problem 2.20b (p. 79) (Just calculate the potential of 2.20b since we haven't covered enough
information to do any other part of this problem yet. The electric field 2.20a is not conservative.)
Answer: The potential for part b is V(x,y,z) = −k(xy²+yz²)
4. Problem 2.21 (p. 82) (Don't calculate the gradient. Do sketch V(r))
Answer: For r>R V=q ⁄ 4πε₀r. For r<R V= (q ⁄ 8πε₀R) (3−r² ⁄ R²)
Hint: Use Gauss' Law to calculate E then use E to calculate V(r).
Read: pp. 13–15, 29–30 (Section(s): 1.2.1, 1.3.3)
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Day # 7: Fundamental theorem of gradients and the gradient of V
Covers: pp. 13–15, 29–30 (Section(s): 1.2.1, 1.3.3)
Goals:
1. Understand and remember the fundamental theorem of gradients.
2. To understand what the gradient means conceptually. (How does gradient relate to ‘slope’?, etc..)
3. To understand the mathematical properties and uses of the gradient.
4. To know and remember the Cartesian form of the gradient.
5. To be able to use the gradient in the Cartesian, spherical, and cylindrical coordinates.
6. To be able to calculate the ‘slope’ or grade of scalar fields in a given direction.
Homework # 7:
1. Answer the following short questions in a sentence or two.
a. What is the equation for the fundamental theorem of the gradient?
b. Line integrals can be different for different paths. In light of that fact does your answer to the
fundamental theorem depend on the path taken in the integral? If not then explain what this means
about the vector field expression ∇ T.
c. What is the geometric interpretation of the the gradient? (Be sure to include the physical meaning of
its magnitude, its direction and how it relates to the grade or ‘slope’ of the scalar field T in a given
direction (for example n̂)
d. Does it bother you that the slope in any arbitrary direction is determined entirely by the slope in one
particular direction? What is the slope in a direction perpendicular to the direction of the gradient?
Does this make sense from your experience with hills? Can you think of a situation where this does
not make sense? (You might want to try thinking about a hill shaped like the corner of a cube tilted
up (in other words a pyramid) is there just one maximal slope direction? is there a direction where
the slope is zero, etc..) What does this say about gradients and differentiability?
e. What does it mean physically if the gradient of a scalar field T is zero at a point? What is meant by
the term stationary point? What are the four types of stationary points? Sketch and label at least 3
of them.
f. What is the Cartesian form of the gradient? Are the cylindrical and spherical forms different in
form?
2. Problem 1.11 a,b (p. 15)
Answer: a. 2xx̂ +3y²ŷ + 4z³ẑ b. 2xy³z⁴x̂ + 3x²y²z⁴ŷ + 4x²y³z³ẑ
Hint: You must show your work.
3. Problem 1.12 (p. 15)
Answer: a. 3 miles N, 2 miles W b. 720 ft c. 311ft/mile northwest
4. Problem 1.13 (p. 15)
5. Problem 1.31a (p. 31)
Answer: 7=7
Read: pp. 83–87 (Section(s): 2.3.3–2.3.4, fig. 2.35)
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Day # 8: Poisson's and Laplace's Equation and the relationship between V and ρ
Covers: pp. 83–87 (Section(s): 2.3.3–2.3.4, fig. 2.35)
Goals:
1. Understand the relationship between V and ρ
2. Know how the Laplacian operator relates to the gradient and divergence
3. Know and remember the Cartesian representation of the Laplacian
4. Know the ‘del’ representation of Laplace's and Poisson's equations
5. Know that Laplace's and Poisson's equations are always second order partial differential equations
6. Know and remember the solution to Poisson's equation for the boundary condition that the potential at
infinity is zero and that all charges are known within that region.
7. Be able to understand and reproduce all relationships between E, V and ρ (See Figure 2.35 on page 87.
Homework # 8:
1. Answer the following short questions in a sentence or two.
a. What is the equation from which ρ can be calculated from V?
b. What is the Cartesian representation of the Laplacian?
c. What are Laplace's and Poisson's equations?
d. What equation allows you to calculate V given ρ if ρ is known everywhere and the potential goes to
zero at infinity?
e. What are all the relationships between ρ, V, and E in electrostatics.
2. Problem 1.25 (p. 24): replace d with Td = rcosθ (work problem in spherical coordinates)
Answer: a. 2 b. −3sin(x)sin(y)sin(z) c. 0 d. 0
Hint: For credit for part d you must do the problem in spherical coordinates.
3. The following refer to the scalar fields (Ta ... Td) of the previous problem.
a. Assuming the Ts represent potentials, what are the charge distributions (with in the reason where V
is represented by T) that produce the potentials?
Answer: ρa = −2ε₀, ρb = 3ε₀sin(x)sin(y)sin(z), ρc = 0, ρd = 0
b. Which of the Potentials of the previous part represent solutions of the Laplace Equation? Of the
Poisson Equation?
c. If there is no charge (in the region where V is valid) for parts c) and d) then how can there be a
potential (and electric field there)? How is it that both of these are solutions to the exact same
equation? (In other words, what property needs to be different when solving the differential
equation to arrive at two completely different solutions?)
d. Show that the boundary conditions necessary to solve Laplace's equation for Vc , Vd (from above) as
well as the case V=C are all different by calculating the Boundary condition necessary for the 2
boundaries z=0 and z=π for these three cases.
Hint: To solve Vd it is easiest to convert it to cartesian coordinates.
Answer: Vc (z=0) = e−5xsin(4y), Vc (z=π) = −e−5xsin(4y), Vd(z=0) = 0, Vd(z=π) = π, VV=C(z=0) = C,
and VV=C(z=π) = C. These 3 sets of B.C are all different. (To be technically correct we still need
to evaluate B.C. bounding the ±x and ±y directions as well.)
4. Problem 2.25 a,b (p. 86) (You don't need to compare the answers to previous problems.)
Answer: a. V = q ⁄ 2πε₀√z²+d² b. V = (λ ⁄ 2πε₀) ln|(L + √L²+z²) ⁄ z|
Hint: When calculating E you can only calculate Ez (unless you use symmetry arguments.) since you
have no clue how V varies in any other direction than along the z-axis. You may use integral tables.
Getting V for b into the simple form of the answer will take a little thought. You will use ln|k²| =
2ln|k|.
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Read: pp. pp. 110–118 (Section(s): 3.1.1–3.1.5)
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Day # 9: Laplace's equation, ‘harmonic functions’, and the Uniqueness Theorem
Covers: pp. pp. 110–118 (Section(s): 3.1.1–3.1.5)
Goals:
1. To review solving differential equations
2. To define the term ‘harmonic functions’ (any possible solution of Laplace's equation.
3. To understand the ‘mean value theorem of harmonic functions’
4. To understand the ‘method of relaxation’
5. To know that harmonic functions tolerate no maximums nor minimums (except on boundary)
6. Know that the solution of any differential equation depends on the boundary conditions
7. Know the boundary conditions that are necessary to solve Poisson's or Laplace's equations
8. To understand the uniqueness theorem for Laplace's (and Poisson's) equations
Homework # 9:
1. Answer the following short questions in a sentence or two.
a. What is a ‘harmonic function’?
b. What is the ‘mean value theorem of harmonic functions’?
c. What is the ‘method of relaxation’? What is it used for?
d. Explain, briefly why harmonic functions tolerate neither maximums nor minimum anywhere except
for the boundary. (Use the ‘mean value theorem of harmonic functions’.)
e. What are the boundary conditions necessary for the solution of Poisson or Laplace equations?
f. What is the uniqueness theorem for solutions of the Laplace and Poisson equations?
2. Answer the following using the properties of ‘harmonic functions’.
a. Problem 3.2 on page 115
b. The potential on the surface in the shape of Mickey Mouse's head is known to be 3.14 Volts
everywhere on that surface. What is the formula for the potential inside the region as a function of
position.
Answer: V = 3.14 V everywhere in the interior
Hint: Uniqueness Theorem!
3. Which of the following are harmonic functions? Find appropriate B.C. for each harmonic function.
a. Ce−nπx ⁄ asin(nπy ⁄ a) (C, n, and a are constants)
b. C ln(s) + k (Cylindrical coordinates: C and k are constants)
c. C[½(3cos²θ−1)]r² (spherical coordinates: C is a constant)
d. Ccos(x)sin(y) (C is a constant)
Answer: a., b., and c. are harmonic.
Hint: For the second part you need to find any appropriate B.C. for each problem. The boundaries
don't have to be the same for each problem. Pick a boundary that is easy to calculate for each
problem.
4. Problem 3.3 on page 116.
Answer: V = −c ⁄ r + k (spherical) and V = cln(s) + k (cylindrical)
Read: pp. pp.127–144 (Section(s): 3.3)
Try to gain an overview of separation of variables. There are a lot of equations in this section. Do not
worry yet, if you don't understand all of them.
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Day # 10: Separation of Variables in Cartesian Coordinates
Covers: pp. pp.127–144 (Section(s): 3.3)
Goals:
1. Understand and be able to use the method of ‘separation of variables’ in Cartesian coordinates.
2. Introduce Fourier series and the Fourier transform.
3. Understand what orthogonal functions are.
Homework # 10:
1. Answer the following short questions in a sentence or two.
a. What are the steps used in the method of ‘separation of variables’?
b. Does separation of variables always work?
c. What is a Fourier series? How is a Fourier expansion similar to a Taylor expansion.
d. If you have two functions F(x) and G(x) what equation would you use to determine if they are
orthogonal.
e. Can two functions be orthogonal for one set of intervals and not another?
f. What is the equation that describes the orthogonality of sin(nπx ⁄ a) with sin(mπx ⁄ a) over the
interval of 0 to a?
g. What is the (finite) Fourier Transform and how does it relate to the Fourier series. For simplicity
only do this for the finite sine Fourier Transform.
2. Problem 3.14 (p. 136)
Answer: a. V(x,y) = ∑Cnsinh(nπx ⁄ a)sin(nπy ⁄ a), where Cn = (2 ⁄ a sinh(nπb ⁄ a))∫Vosin(nπy ⁄ a)dy b.
V(x,y) = (4Vo ⁄ π)∑sinh(nπx ⁄ a)sin(nπy ⁄ a) ⁄ (n sinh(nπb ⁄ a), where the sum is only over odd n.
Hint: This problem involves a fair amount of work if it is done right. To obtain full credit you must
show me all of the steps.
Read: pp. 90–96 (Section(s): 2.4)
15
Day # 11: Energy of formation for a charge distribution.
Covers: pp. 90–96 (Section(s): 2.4)
Goals:
1. Be able to calculate the energy it takes to create a charge distribution of point charges by bringing in
each charge one at a time from infinity.
2. Know and remember the equation for determining the energy of formation for point charges.
3. Understand where the factor of ½ in the equation for the formation energy comes from.
4. Know and remember the equation for determining the energy of formation for distributed charges in
terms of ρ and V.
5. Be able to calculate the energy of formation for simple charge distributions.
6. Know and remember the equation for determining the energy of formation for charge distributions in
terms of the electric field E
Homework # 11:
1. Answer the following short questions in a sentence or two.
a. What is the equation for the energy of formation of point charges in terms of V and qi?
b. What is the equation for the energy of formation of a continuous charge distribution in terms of V
and ρ?
c. What is the equation for the energy of formation of a continuous charge distribution in terms of the
electric field E?
d. Which is more correct, to say that the energy is stored in the Electric field or that the energy is
stored in the charges with the potentials of the other charges??
2. Problem 2.31 (p. 93) do part b. 2 different ways (using 2.42 and by adding up the energy it takes to
bring in each charge separately.)
Answer: a. (q² ⁄ 4πε₀a)(−2 + 1 ⁄ √2) b. 2(q² ⁄ 4πε₀a)(−2+1 ⁄ √2)
Hint: Remember V in equation 2.42 does not include the potential due to the charge you are
measuring the potential at. Use symmetry while using 2.42 to simplify the problem.
3. Problem 2.32 (p.95)
Answer: (1 ⁄ 4πε₀)3q² ⁄ 5R
Read: pp. pp. 202–214 (Section(s): 5.1)
16
Day # 12: Lorentz Force Law, Currents and Current Densities
Covers: pp. pp. 202–214 (Section(s): 5.1)
Goals:
1. Understand and remember the Lorentz Force Law for point charges moving at speed v
2. Understand and remember the definitions for the current I, surface current density K and bulk current
density J
3. Understand and remember how calculate I from K or J and vice versa.
4. Understand and remember how how I, K, and J depend on λ, σ, and ρ respectively.
5. Understand and remember the Lorentz force Law on currents in terms of I (1D), K(2D) and J(3D).
Homework # 12:
1. Answer the following short questions in a sentence or two.
a. What is the Lorentz Force Law for a point charge (charge = Q) moving at a velocity v in an electric
field E and a magnetic field B?
b. True or False: According to the Lorentz Force Law, magnetic fields can do no work.
c. What are the definitions of current I, sheet current density K and bulk current density J
d. What equation allows you to calculate I through a given surface from J through that surface.
e. What are the equations relating I, K, and J to λ, σ, and ρ respectively?
f. What is the continuity equation and what does it express.
g. What are the representations of the Lorentz Force Law in terms of I, K, and J.
2. Problem 5.3 (p. 208)
3. Problem 5.4 (p. 214)
Answer: F = Ika² ẑ̂
4. Problem 5.5 (p. 214)
5. Problem 5.6 (p. 214)
Read: pp. pp.215–219 (Section(s): 5.2)
17
Day # 13: Biot-Savart Law and Currents as ‘sources’ for B
Covers: pp. pp.215–219 (Section(s): 5.2)
Goals:
1. Know what condition(s) are neccesary for ‘magnetostatics’
2. Know and remember the Biot-Savart Law
3. Know what condition is necessary for the Biot-Savart law to be valid.
4. Know the definition for the ‘permeability of free-space’ µ₀ and the units used to measure B.
5. Know that unlike E the field lines of B only comes in loops. (There are no magnetic charges or
‘magnetic monopoles’.
6. Know and understand that ∇ ·B = 0 for all B (magnetostatic and otherwise).
7. Know that current (or moving charge) are ‘circulation’ sources of B.
8. Given a ‘magnetic field line’ diagram be able to determine if the diagram represents a valid magnetic
field and be able to determine where there must be current.
Homework # 13:
1. Answer the following short questions in a sentence or two.
a. What condition is necessary for a system to be magnetostatic? Is a point charge moving at a
constant velocity an example of magnetostatics? Why or why not?
b. Write down the Biot-Savart Law.
c. What condition is necessary for the Biot-Savart Law to be exactly true? In practice the Biot-Savart
Law is often used when this condition is violated. How can that be?
d. What is the constant µ₀ known as? What are the SI and cgs units of magnetic field? What is the
conversion factor between these two units? Relate the magnitudes of these two units with the
magnetic fields around us.
e. What is the value of ∇ ·B for magnetostatics? Is the same true for systems that are not
magnetostatic? Explain why for both cases.
f. In electrostatics charges are sources of electric field lines, while electric field lines begin and end
only at charges. What physical quantity is the ‘source’ of magnetic field lines in magnetostatics?
How does the magnetic field lines relate to its source?
g. Place an x across all regions in the following diagram that are not possible for magnetic field lines.
Circle all regions where there is current, assuming that the diagram represents magnetic field lines.
Indicate the direction of the current. (Assume that there is symmetry along the z-axis going in and
out of the paper such that there is no change in the magnitude nor direction of the magnetic field
when you move in that direction.)
2. Fill in the missing steps between equations 5.36 and 5.37 in the text (p. 217). Be sure to determine the
direction of the force per unit length as well.
3. Problem 5.8 (p. 219)
Answer: a. B = √2µ₀I ⁄ πR b. B = nµ₀Isin(π ⁄ n) ⁄ 2πR
18
4. Problem 5.11 (p. 220)
Answer: B = ½ µ₀nI(cosθ₂−cosθ₁)
5. Problem 5.12 (p. 220)
Answer: v = c
Hint: plug in values for ε₀ and µ₀ to get the answer. The answer is exactly c and is not an accident.
Read: pp. pp. 225–230 (Section(s): 5.3.3)
19
Day # 14: The integral form of Ampere's Law
Covers: pp. pp. 225–230 (Section(s): 5.3.3)
Goals:
1. Understand and remember Ampere's Law in integral form
2. Know what conditions are necessary for Ampere's law to be true
3. Be able to use Ampere's Law to calculate B for simple symettric cases
Homework # 14:
1. Answer the following short questions in a sentence or two.
a. What is the integral form of Ampere's Law?
b. What is meant by the term Ienc in Ampere's Law? In particular explain what is enclosing Ienc and how
does one determine the direction of Ienc . What would happen if the line integral of Ampere's Law
was integrated in the other direction?
c. Is Ampere's Law valid just for magnetostatics or always? How does this compare to Gauss' Law?
d. Is Ampere's Law valid only for highly symettric cases or is is valid for all magnetostatic cases.
e. What 3 symettrical systems can be easily exploited to calculate B from Ampere's Law?
2. Problem 5.13(p. 231)
Answer: a. B = 0 for s<a and µ₀Iφ̂ ⁄ 2πs for s>a
b. B = µ₀Is²φ̂ ⁄ 2πa³ for s<a and µ₀Iφ̂ ⁄ 2πs for s>a
3. Problem 5.14(p. 231)
Answer: B = −µ₀Jzŷ̂ inside the slab, and ∓µ₀Jaŷ̂ outside the slab where the minus sign if for z>a and
the plus sign is for z<−a.
4. Problem 5.16(p. 231)
Answer: a. B = µ₀σv, into page between the plates and zero elsewhere.
b. fm = µ₀σ²v² ⁄ 2, up.
c. fe = σ² ⁄ 2ε₀ , down and v must = c for the two forces to balance.
5. Investigate Problem 5.18 in the following manner. Imagine an infinite wire of diameter D coaxial with
the z axis such that J is uniform = J₀ẑ̂. We are interested in calculating the current enclosed by a circle
of diameter D centered on the z axis and in the xy-plane. Calculate the current through the following
three surfaces: a solid circle in the xy-plane that is bounded by our enclosing circle, the curved portion
of the outside of a hemisphere of diameter D bounded by our enclosing circle with the curved side up,
and finally the curved and top surfaces of a right cylinder (diameter D and height L) such that the
bottom side of the curved portion of D is the enclosing circle. Perform the same calculations assuming
the the current stops just above the xy-plane. What is the difference between the two cases? What does
this say about when Ampere's Law is valid?
Answer: For the case of the uniform current Ienc = πJoD² ⁄ 4 for all three surfaces. For the case where
the current suddenly stops the flat surface yields the same results but the other two surfaces leads to
Ienc being zero.
Hint: Note that Ienc is the same for all surfaces only in the first case when ∇ ·J = 0.
Read: pp. pp. 34–36, pp. 19–20, pp.222–224 (Section(s): 1.3.5, 1.2.5, 5.3.1)
The fundamental theorem of curls and the differential form of Ampere's Law. Note that I have reversed the
order of the first two sections. The reason is that the fundament theorem of the curl in section 1.3.5 defines
the curl and is the best way to understand the curl. On the other hand section 1.2.5 only gives the cartesian
representation of the curl which doesn't help get a physical intuition of what the curl means.
20
Day # 15: The Curl, fundamental theorem of curl, and differential form of
Ampere's Law
Covers: pp. pp. 34–36, pp. 19–20, pp.222–224 (Section(s): 1.3.5, 1.2.5, 5.3.1)
Goals:
1. Understand and remember the fundamental theorem of the Curl (Stokes Theorem)
2. Remember the Cartesian representation of the Curl
3. Be able to calculate the curl of a vector field in cartesian, spherical, and cylindrical coordinates
4. Know and remember the differential form of Ampere's Law
5. Know and remember the value of the second derivatives of the form ∇ ·∇
∇ ×A and ∇ ×∇
∇V
6. Know the relationship between the curl of a vector field A and whether or not that vector field is
conservative.
Homework # 15:
1. Answer the following short questions in a sentence or two.
a. Write down the fundamental theorem of the Curl.
b. How does the area of the area integral on the left hand side of Stoke's Law relate to the closed loop
over which the the line integral on the right side of the equation integrates. (Make sure to include
how the directions relate as well.)
c. What is the cartesian representation of the curl?
d. Do other coordinate systems have the same representation for the curl as the cartesian coordinate
system? The notation ∇ ×A is suggestive that you can find some ∇ different then the one used for
Cartesian coordinates where the curl{A} = ∇ cross A. Is this possible for non Cartesian
coordinates?
e. What is the differential form of Ampere's Law (for magnetostatics)? Is this form of Ampere's Law
valid for all cases including magneto and electro dynamics?
f. What are the values for the second derivatives of the form ∇ ·∇
∇ ×A and ∇ ×∇
∇V
g. How would you determine if a vector field A is conservative using nothing but ∇ ×A? Prove this
relationship by using Stoke's theorem.
2. Problem 1.18(p. 20) If these vector fields represent B then what are the current densities J that created
each?
Answer: a. − 6xz x̂̂ + 2z ŷ̂ + 3z² ẑ̂
b. − 2y x̂̂ − 3z ŷ̂ − x ẑ̂
c. 0
to determine the current densities for each part just multiply by µ₀
3. Problem 1.33(p. 36)
Answer: − 8 ⁄ 3 = − 8 ⁄ 3
4. Problem 1.34(p. 36)
Answer: 4 ⁄ 3 = 4 ⁄ 3
Read: pp. pp.232–240 (including Figure 5.48 on page 240) (Section(s): 5.3.4–5.4.1 (and Figure 5.48))
21
Day # 16: The vector potential A
Covers: pp. pp.232–240 (including Figure 5.48 on page 240) (Section(s): 5.3.4–5.4.1 (and Figure 5.48))
Goals:
1. Know and understand the relationship between the vector potential A and the magnetic field B.
2. Be able to calculate the magnetic field B from the vector potential A
3. Be able to use Ampere's Law in certain high symmetry cases to calculate A from B
4. Understand the physical nature of A (as a ‘potential momentum’)
5. Know and understand what is meant by A being gauge invariant
6. Know what the Coulomb gauge is (∇
∇ ·A = 0)
7. Know and understand the relationship between A and J in the Coulomb gauge
Homework # 16:
1. Answer the following short questions in a sentence or two.
a. What is the relationship between the vector potential A and the magnetic field B it corresponds to?
b. Of what physical property of nature is A a measure?
c. What is meant by A being ‘gauge invariant’
d. What is the equation for the Coulomb gauge?
e. In the Coulomb gauge, how does one calculate J from A? A from J?
f. Show that A has units of momentum per unit charge.
2. Problem 5.23(p. 239)
Answer: kφ̂̂ ⁄ µ₀s²
3. Problem 5.24(p. 239)
Hint: You may use the Product Rules table at the front of the book to simplify this problem slightly.
4. Problem 5.25(p. 239)
Answer: Aout = −(µ₀ ⁄ 2π)Iln(s ⁄ R)ẑ̂, Ain = −(µ₀ ⁄ 4πR²)(s² − R²)
Hint: ‘By whatever means you could think of’ probably means ‘see example 5.12’ in this case. First
use the symmetry of B to show that A = A(s)ẑ̂ (make sure to explain your reasoning here). Then
arbitrarily assume that A(R) = 0 to simplify the problem.
5. Lets examine the results of Example 5.12(p.238) by answering the following questions about that
example. What is B outside the cylinder? What is A outside the cylinder? Let us switch to a new gauge
Ang such that Ang = A + ∇ λ, such that ∇²λ = f(r) where λ represents an arbitrary solution of the Poisson
equation with f(r) being an arbitrary function. What are the values of ∇ ×Ang and ∇ ·Ang? If the current
is reduced to zero after a time of to seconds what is the value of Ang outside the cylinder at this time?
What is the value of ∆Ang ≡ Ang(t = to) − Ang(t = 0)? Does ∆Ang depend on the gauge? Since ∆Ang
represents a change in ‘potential momentum’ what is the momentum change of a particle having charge
q outside of a solenoid when the current is turned off in to seconds? What is the average force over that
same time interval on that charge?
Hint: This problem will be explored in more detail when we get to electrodynamics. The purpose of
this problem is to give you a better feel for what A means physically. Your final answer to this
problem represents a real force that is very important to electrodynamics and allows us to generate
electricity. You may need to review the relationship between force and momentum for the last part
of this question.
Read: pp. 285–293 (Section(s): 7.1.1–7.1.2)
22
Day # 17: Ohm's Law and the ‘Electromotive Force’
Covers: pp. 285–293 (Section(s): 7.1.1–7.1.2)
Goals:
1. To know and understand ‘Ohm's Law’
2. To have a qualitative understanding of the origins of Ohm's Law
3. To be able to calculate the total current I going through a surface given E
4. To be able to calculate how the Resistance of an object depends on the conductivity of a material and
the geometry of the object (including how the Voltage is applied) for simple geometries.
5. To understand and remember the Joule heating law
6. To know and remember the definition of the EMF (ℇ) on a loop due to a source f
7. To know that the EMF due to an Electrostatic field is always zero for all loops
Homework # 17:
1. Answer the following short questions in a sentence or two.
a. What is Ohm's Law in terms of J and f where f represents a force per unit charge? In terms of J and
E? In terms of V and I?
b. Is Ohm's law always valid? When is it valid?
c. What is the Joule heating law? Where does the power come from what is it being converted to?
d. What is the definition of the EMF ℇ of a source force per unit charge f around a loop?
e. What is the EMF of an electrostatic E field over any loop?
2. Problem 7.1 (p. 290)
Answer: a. I = 4πσ(Va−Vb) ⁄ (1 ⁄ a − 1 ⁄ b)
b. R = (1 ⁄ 4πσ)(1 ⁄ a − 1 ⁄ b)
c. I = 2πσaV
Hint: You will need to assume the inner sphere has a charge Q, then use Gauss' Law to calculate E.
From E you can calculate both I and V as a function of Q. Eliminate Q to get V as a function of I.
Assume the permittivity of the conducting medium is ε₀.
3. Problem 7.5 (p. 293)
Answer: R = r
Hint: First determine the power dissipated by the load resistance. Assume Joule heating. Then
maximize this expression (using the same technique for minimizing). Remember that the process for
minimizing doesn't actually produce a minimum but a maximum, a minimum or an inflexion point.
Read: pp. 294–309 (Section(s): 7.1.3–7.2.1)
23
Day # 18: Motional EMF and Faraday's Law
Covers: pp. 294–309 (Section(s): 7.1.3–7.2.1)
Goals:
1. Understand how the motion of a conducting loop generates a motional emf
2. Be able to calculate the motional emf of a moving loop
3. Understand that ‘work done per unit charge’ and emf are two separate concepts.
4. Know and remember the ‘flux rule’ for emf
5. Know when the ‘flux rule’ is valid
6. Know and remember Faraday's law
7. Know when Faraday's law is valid
8. Be able to calculate the electric field due to a changing magnetic field in simple situations
9. Be able to use Lenz's law to determine the direction of the current in a loop with a changing magnetict
field
Homework # 18:
1. Answer the following short questions in a sentence or two.
a. A rectangular loop is pulled by hand with a steady speed v through a static magnetic field such that
its leading edge is in an region having zero magnetic field. The result is to turn mechanical energy of
the pull into electrical energy. What force does the work on the charges in the wire? What force
causes the emf on the wire? What path is integrated over to obtain the emf? What path is is
integrated over to obtain the work? Does this agree with the idea that magnetic fields can do no
work?
b. What is the definition of magnetic flux?
c. What is the equation used to determine the emf generated by a moving loop in a magnetic field B in
terms of the magnetic flux?
d. When is the ‘flux rule’ valid? (Magnetostatics only, Electrodynamics only, or always)
e. What is Faraday's law and when is it valid? Does this agree with the corresponding equation for
electrostatics? Explain.
2. Problem 7.8 (p. 300)
Answer: φ = (µ₀Ia ⁄ 2π)ln((s+a) ⁄ s)
ℇ = µ₀Ia²v ⁄ 2πs(s+a) counterclockwise
ℇ=0
3. Problem 7.12 (p. 305)
Answer: (πa²ω ⁄ 4R) bosin(ωt)
Read: pp. 310–320 (Section(s): 7.2.3–7.2.4)
24
Day # 19: Inductance and the energy stored in magnetic fields
Covers: pp. 310–320 (Section(s): 7.2.3–7.2.4)
Goals:
1. Know the definitions of mutual inductance M, and self inductance L
2. Know the equations that relate the induced emf due in one loop due to the current in another loop and
to the current in that loops as well
3. Know the unit of inductance
4. Know what is meant by back emf
5. Know the equation that decribes the work it takes to create a current I in a coil having self inductance
L.
6. Know the equation that descibes the work it takes to create a current distribution J in terms of the
magnetic fields it creates
7. Be able to calculate mutual and self inductances of simple current geometries
8. Be able to calculate the energy of creation for simple current geometries in terms of L and I and also in
terms of B
Homework # 19:
1. Answer the following short questions in a sentence or two.
a. What are the definitions of mutual inductance M and self inductance L?
b. What equation allows the calculation of the emf induced in a second loop due to a changing current
in the first loop?
c. What equations allows the calculation of the emf induced in a loop by a changing current in that
loop.
d. What is the SI unit of inductance?
e. Define the term ‘back emf’
f. What is the equation that decribes the work it takes to create a current I in a coil having self
inductance L?
g. What is the equation that descibes the work it takes to create a current distribution J in terms of the
magnetic fields it creates
2. Problem 7.22 (p. 316)
Answer: L ⁄ l = µ₀n²πR²
Hint: If this takes more then 2 lines you are probably doing it the hard way
3. Problem 7.24 (p. 316) (Calculate M as an intermediate step.)
Answer: M = 1.38×10⁻⁶H
ℇ = 2.61×10⁻⁴ V sin(ωt)
Ir = 5.22×10⁻⁷ A sin(ωt)
ℇback = −2.74×10⁻⁷ V cos(ωt)
ratio = 1.05×10⁻³
4. Problem 7.26 a. and c. (p. 320)
Answer: W ⁄ l = ½µ₀n²πR²I²
Hint: You may use the results of problem 7.22
Read: pp. 321–327 (Section(s): 7.3.1–7.3.3)
25
Day # 20: The Displacement Current and Maxwell's equations
Covers: pp. 321–327 (Section(s): 7.3.1–7.3.3)
Goals:
1. Know how Ampere's law is invalid for dynamics
2. Know and remember Maxwell's correction to Ampere's law
3. Know the definition of displacement current
4. Remember Maxwell's equations
5. Be able to show that a given set of E and B fields satisfy Maxwell's equations
Homework # 20:
1. Answer the following short questions in a sentence or two.
a. Explain why Ampere's law is invalid for dynamics (when ∇ ·J ≠ 0)
b. How did Maxwell fix Ampere's Law for the case were ∇ ·J ≠ 0?
c. What is the definition of displacement current.
d. Write down all four of Maxwell's equations.
2. Problem 7.33 (p. 325)
Answer: Jd = (µ₀ε₀ ⁄ 2π)ω²I ln(a ⁄ s)ẑ
Id = µ₀ε₀ω²Ia² ⁄ 4
Id ⁄ I = (ωa ⁄ 2c)² ⇒ ω = 6 × 10¹⁰ Hz
3. Problem 7.34 (p. 327)
Answer: ρ = −qδ³(r)θ(t) + (q ⁄ 4πr²)δ(vt−r)
Hint: See section 1.5 in text (in particular Equation 1.100 on page 50) and problem 1.45b on page 49.
Read: pp. 96–106 (Section(s): 2.5)
We have established what happens with E and B where we control all the charges and currents physically
and directly. In real life though we only control a portion of the charge since all materials are made out of a
large number of positive and negative charges that are pushed around by the electric and magnetic fields
we create. These materials respond by having charge move around to produce their own charge
distributions and current distributions which in turn produces their own electric and magnetic fields. The
final third of this course is concerned with developing Maxwell's equations for ‘matter’ which takes this
into account. We will begin this section with the study of what happens when an electric field is applied to
a conductor.
26
Day # 21: Conductors and Capacitance
Covers: pp. 96–106 (Section(s): 2.5)
Goals:
1. To gain insight into what happens when an electric field is applied to a material by examining first what
happens when an electric field is applied to a conductor
2. To understand that a conductor is a material with a large amount of charge that is free to move in the
material
3. To know and remember that five properties of a conductor in static equilibrium
4. To know the relationship between the surface charge densitiy of a conductor and the local electric field
just outside of the conductor.
5. To understand that the surface of conductors feel an outward pressure into the electric field and be able
to calculate its magnitude
6. To know the definition of capacitance
7. To be able to calculate the energy stored in a capacitor in terms of C and V.
Homework # 21:
1. Answer the following short questions in a sentence or two.
a. What condition is necessary for a conductor to be in static equilibrium.
b. What are the five properties of a conductor in electrostatic equilibrium.
c. What is the mathematical relationship between the Electric field just outside of a conductor and the
local surface charge density σ?
d. What is the definition of capacitance in terms of V and Q, where V is the voltage difference
between two conductors (one of which will be charged to +Q and the other to −Q by the potential
difference) and Q is the charge created by the potential difference V?
e. What is the energy stored in a capacitor in terms of C and V?
2. Problem 2.36 (p. 101)
Answer: a. σa = −qa ⁄ 4πa²; σb = −qb ⁄ 4πb²; σR = (qa+qb) ⁄ 4πR²
b. Eout = 1 ⁄ 4πε₀ (qa+qb) ⁄ r² r̂̂
c. Ea = qar̂̂a ⁄ 4πra²; Eb = qbr̂̂b ⁄ 4πrb²
d. 0
e. only σr and Eout changes
3. Problem 2.37 (p. 103)
Answer: P = Q² ⁄ 2ε₀A²
4. Problem 2.39 (p. 106)
Answer: C ⁄ l = 2πε₀ ⁄ ln(b ⁄ a)
Read: pp. p. 160, 146–154 (Section(s): 4.1.1, 3.4)
27
Day # 22: Polarization and the Multipole Expansion
Covers: pp. p. 160, 146–154 (Section(s): 4.1.1, 3.4)
Goals:
1. Qualitatively understand how an applied Electric field affects an insulating material in Electrostatics
2. Understand the difference between ‘free’ and bound charge
3. Qualitatively understand electric polarization including the two different ways that an electric field can
cause an insulating material to become polarized
4. Know qualitatively what the multipole expansion is and remember the names and features of the first 3
terms.
5. Be able to use the multipole expansion to determine a one term approximation for the potential far
away from a charge distribution
6. Know and remember the potential for the monopole and the dipole terms.
7. Know and remember the definition of the dipole moment for both continuous and discrete charges
8. Know the dipole moment (its magnitude and direction) of two equal and opposite charges separated by
a distance d.
9. Be able to sketch the electric field of a dipole.
Homework # 22:
1. Answer the following short questions in a sentence or two.
a. Explain qualitatively what happens when an electric field is applied to an insulating material. (Make
sure to explain about any charges that are produced and the electric fields that these bound charges
themselves create.) How does this ‘induced’ electric field compare to the applied electric field in
magnitude and direction?
b. What is the difference between ‘free’ and ‘bound’ charge?
c. What is meant by an atom or a material being electrically ‘polarized’? polarized materials (or atoms)
produce an electric field even though the total charge is zero.
d. True or False: some molecules are naturally polarized even in the absence of an electric field. If
true, then explain how material made up of these molecules is not itself polarized.
e. What are the two ways that an applied electric field can cause a non-polarized insulating material to
become polarized?
f. Write down the potentials for the monopole and the dipole terms in the multipole expansion.
g. Explain why for large distances the multipole expansion may be approximated by the value of the
first non-zero term in the expansion. (Remember that large distances means r is much greater then
r'.)
h. What is the definition of the dipole moment both for a continuous charge density and discrete
charges?
i. What is the dipole moment of two equal and opposite charges ±q separated by a distance d (sketch
the charges and the dipole moment to show the direction of p).
j. Sketch the electric field of a dipole moment.
2. Problem 3.27 (p. 151)
Answer: V = 2qacos(θ) ⁄ 4πε₀r²
3. Problem 3.28 (p. 151)
Answer: a. p = 4πR³kẑ̂ ⁄ 3
b. V = kR³cos(θ) ⁄ 3ε₀r²
Read: pp. 160–173 (Section(s): 4.1–4.2)
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Day # 23: Polarization and bound charge
Covers: pp. 160–173 (Section(s): 4.1–4.2)
Goals:
1. Know and remember the definition of the polarization (P) field
2. Know and remember the relationships between P and the surface (σb) and bulk (ρb) bound charge
densities.
3. Given a Polarization field P be able to calculate ρb and σb
4. Know that P ⁄ ε₀ has same units as E but it is not the electric field of the bound charges
5. Know P has same outflow source (divergence) as Ebound but has a non-zero circulation source (∇
∇ ×P ≠ 0)
Homework # 23:
1. Answer the following short questions in a sentence or two.
a. What is the definition of the Polarization field P in terms of the total dipole moment of a very small
volume?
b. Given the polarization field P what are the equations that allow you to calculate the bound surface
charge density σb and bulk charge density ρb?
c. Which of the following statements are true for the Polarization of a dielectric? For the electric field
of an (electrostatic) bound charge distribution Eb? For both? for neither?
i. has a non-zero curl
ii. has a non-zero divergence
iii. its divergence = −ρb
iv. its divergence = ρb ⁄ ε₀
v. exists only in the material
vi. exists outside the material as well as in
vii. no portion of it forms loops
2. Problem 4.10 on p. 169
Hint: This is much simpler then example 4.2 since P is in the r̂̂ direction. you will probably want to
use Gauss' law.
3. Problem 4.11 on p. 170 (In addition I want you to show that all the relations you explored in the last
part of the first problem of this homework is true. In particular note where the curl of P is not zero.)
4. Problem 4.14 on p. 173
Hint: The total charge is the volume integral of the bulk charge plus the surface integral of the surface
charge. (This is a one (maybe two) liner.)
Read: pp. 175–184 (Section(s): 4.3–4.4.1)
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Day # 24: The displacement field D and linear dielectrics
Covers: pp. 175–184 (Section(s): 4.3–4.4.1)
Goals:
1. To become familiar with the displacement field D
2. To know the two sources of D (outflow source = ρf and circulation source = ∇ ×P)
3. To become familiar with linear dielectrics
4. To be able to solve for the electric field in the presence of dielectrics for highly symmetric cases using
Guass' law for D
Homework # 24:
1. Answer the following short questions in a sentence or two.
a. What is the definition of the displacement field D?
b. What are the values for the curl and the divergence of D?
c. What are the definitions of electric susceptibility (χe), permittivity (ε), and relative permittivity
(otherwise known as the dielectric constant εr)
d. Answer the following statements as either true or false:
All materials are linear dielectrics
Many materials are linear dielectrics for small E
In order for a material to be a linear dielectric E and P have to point in the same direction.
The most general form of a linear dielectic is P = ε₀χe ·E, where χe is the susceptibility tensor.
In the (increasingly important) field of non-linear optics many materials have a cubic term in E for
strong E
χe, εr, and ε are frequency independent for oscilatting E
There is no Coulomb's Law for D (even though there is a Guass' law) because ∇ ×D ≠ 0.
2. Problem 4.15 on p. 177
Answer: E = −(k ⁄ ε₀r)r̂̂ in the shell (a<r<b) and E = 0 everywhere else
3. Problem 4.17 on p. 179
4. Problem 4.18a–d on p. 184
Answer: a. D = σ down between the plates and zero elsewhere
b. E1 = σ ⁄ 2ε₀ down, E2 = 2σ ⁄ 3ε₀ down
c. P1 = σ ⁄ 2 down; P2 = σ ⁄ 3 down
d. V = 7σa ⁄ 6ε₀
Read: pp. 160–165, 191–196 (Section(s): 4.1, 4.4.3–4.4.4)
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Day # 25: Energy in Dielectric Systems
Covers: pp. 160–165, 191–196 (Section(s): 4.1, 4.4.3–4.4.4)
Goals:
1. To understand and remember the equation for the energy of formation of an electic field in the
presence of a linear dielectric
2. To understand and remember the equation for the work done on dipoles by the electric field
3. To rememember and be able to use the equations relating the torque and force on an electric dipole
moment due in an electric field.
Homework # 25:
1. Answer the following short questions in a sentence or two.
a. What is the equation in terms of the (electric field and the displacement field) for the energy of
formation of a free charge in the presence of a linear dielectric(s)?
b. Is the above equation valid for non-linear dielectrics?
c. What is the equation for the energy of formation of a free charge in the presence of a dielectric
expressed in terms of the electric field E and the polarization P? (Show both the general form of the
equation and the simplification for linear dielectrics.)
d. What are the equations that allow you to calculate the torque (N) and force (F) on an electric
dipole (p)?
e. What is the equation for the energy of an ideal electric dipole in the presence of an electric field?
2. Problem 4.26 on p. 193
Answer: (Q² ⁄ 8πε₀(1+χe)) (1 ⁄ a + χe ⁄ b)
3. Problem 4.7 on p. 165
Hint: Bring in a dipole from infinity then rotate it. To simplify the problem bring in the dipole such
that it is perpendicular to the electric field (see footnote on p. 165).
4. Problem 4.9 on p. 165
Answer: See equations 3.103 and 3.104 for the answer to part b. You will show that the answer for
part a. is equal and opposite to that of b.
Hint: This is not mathematically trivial. Solve the problem in cartesian coordinates with the charge q
located at the origin. It may be a little easier to calculate the x-component of F then use symmetry
to deduce the form of the other components.
Read: pp. pp. 242–246, pp. 255–262 (Section(s): 5.4.3, 6.1)
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Day # 26: The magnetic dipole moment m and magnetization M
Covers: pp. pp. 242–246, pp. 255–263 (Section(s): 5.4.3, 6.1)
Goals:
1. Have at least a cursorary understanding of the multipole expansion of the vector potential
2. To remember and be able to calculate the magnetic dipole moment m
3. To know, remember, and be able to calculate the vector potential Adip of a magnetic dipole (for an ideal
dipole)
4. To be able to sketch the magnetic field of a magnetic dipole and to know how it is similar and different
to the sketch of the electric field of an electric dipole
5. To be able to calculate the torque and force on an ideal magnetic dipole m in the presence of a
magnetic field B
6. To understand the properties of paramagnetic, diamagnetic, and ferromagnetic materials
7. To know the definition of magnetization.
Homework # 26:
1. Answer the following short questions in a sentence or two.
a. In the multipole expansion of the vector potential, what is the value of the monopole term?
b. What is the definition of the magnetic dipole moment of a loop of current I and area a where the
area is in a plane and the direction of a is normal to that plane.
c. What is the dipole contribution Adip to the multipole expansion of the vector potential A due to the
dipole moment m?
d. Sketch both the magnetic field of a magnetic dipole and the electric field of an electric dipole from
a perspective that close enough to the dipoles to see the fields inside the dipole. Show where the
fields are the same and show where they are different.
e. What are the equations used to calculate the torque N and the force F on a magnetic dipole m in the
presence of a magnetic field B?
f. What are the properties of diamagnetic, paramagnetic, and ferromagnetic materials?
g. What is the definition of magnetization M in terms of the dipole moment m of a tiny volume (∆τ)?
2. Problem 5.34 on p. 246
Answer: a. m = IπR²ẑ̂
b. B ≈ (µ₀ ⁄ 4πr³) IπR²(2cosθ r̂̂ + sinθ θ̂̂)
3. Problem 6.1 on p. 259
Answer: N = − (µ₀ ⁄ 4r³) (abI)² x̂̂
4. Problem 6.6 on page 263
Hint: You will need to find a periodic table for this
Read: pp. 263–277 (Section(s): 6.2–6.4)
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Day # 27: Bound Current and the H field
Covers: pp. (Section(s): )
Goals:
1. To understand the physical origins of the surface Kb and bulk Jb bound current densities
2. To know and remember the relationships between the Magnetisation M and the bound current
densities, Kb and Kb
3. To know and remember the definition of the ‘H’ field
4. To know and remember ‘Ampere's law for H’
5. To be able to use ‘Ampere's law for H’ to calculate H and B for certain high symmetry cases
6. To know and remember the definitions for magnetic susceptibility χm and permeability µ
Homework # 27:
1. Answer the following short questions in a sentence or two.
a. Write down the equations that allow you to calculate the bound current densities Kb and Jb from the
magnetization M
b. What is the definition of the H field?
c. What is ‘Ampere's Law for the H field’?
d. True or False: The divergence of the H field like the magnetic field B is always zero.
e. What is the value for the divergence of H in terms of M?
f. What are the definitions of the magnetic susceptibility χm and permeability µ?
g. Are ferromagnetic material linear media in magnetization?
2. Problem 6.8 on p. 265 (Do this by calculating the bound currents then using Ampere's Law)
Answer: B = µ₀M (inside)
B = 0 (outside)
Hint: The answer suggests that using H and its ‘Ampere's Law’ the answer would come even faster.
(Since there are no free currents and because ∇ ·H = −∇
∇ ·M which for this case = 0 we can write
that H = 0. The answer then falls in our lap.)
3. Problem 6.12 on p. 116
Answer: B = µ₀M (inside)
B = 0 (outside)
Hint: See hint to the previous problem
4. Problem 6.16 on p. 277
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