Description
1) Consider a sphere of radius 𝑅 and total charge 𝑄 that has been embedded with an r‐dependent
charge density:
𝜌ሺ𝑟⃗ሻ ൌ 𝐶 𝑟
a) Write and solve an integral to determine 𝐶 in terms of the properties of the sphere, 𝑄 and 𝑅.
For the rest of the problem, use this result for 𝐶 in 𝜌ሺ𝑟⃗ሻ.
b) Explain why and how you can use Gauss’ Law to solve for the electric field of the charged
sphere.
c) Set up your solution and determine 𝐸ሬ⃗ሺ𝑟⃗ሻ for all 𝑟. You will have somewhat different results for
0 𝑟𝑅 and for 𝑟𝑅. These results should agree for 𝑟ൌ𝑅.
d) Solve for 𝜙ሺ𝑟⃗ሻ for all r. As usual, define 𝜙ሺ∞ሻ ൌ 0.
e) Solve for the total electric potential energy of the charged sphere. This can be considered the
total energy (work) necessary to bring all the charges together in the sphere.
f) Consider a similar sphere with the charge density 𝜌ሺ𝑟⃗ሻ ൌ 𝐶 𝑟 cosଶ 𝜃. Set up a multipole
expansion to solve for 𝜙ሺ𝑟⃗ሻ and show (explain) that there will only be two terms in the
expansion. If you wish, you can solve this.
2) The Magnetic analog to the problem above is a long, current‐carrying
wire with a current density that varies across the radius of the wire.
Use polar coordinates: 𝑧̂along the wire, 𝑟̂the radial direction
perpendicular to 𝑧̂, and 𝜙 the azimuthal angle around 𝑧̂.
The wire has a radius 𝑅 and total current 𝐼 in the 𝑧̂direction. The current density is:
𝐽⃗ሺ𝑟⃗ሻ ൌ 𝐶 𝑟 𝑧̂
a) Write an equation relating the total current in the wire, 𝐼, to the current density, 𝐽⃗ሺ𝑟⃗ሻ for the
wire. Use this to Derive an expression for the constant 𝐶 in terms of properties of the wire, 𝐼
and 𝑅.
b) Use the symmetry of the problem and the relation between the current (density) and the
magnetic field to determine the general form for the magnetic field, 𝐵ሬ⃗ሺ𝑟⃗ሻ. Specifically, what
direction is the field and how does it depend on 𝑟,𝜙, and 𝑧?
c) Explain why you can use Ampere’s Law to solve for 𝐵ሬ⃗ሺ𝑟⃗ሻ.
d) Calculate 𝐵ሬ⃗ሺ𝑟⃗ሻ everywhere. Be sure to explain your approach.
e) Show that the magnetic potential energy of the wire is infinite. This shouldn’t be surprising, as it
is an infinitely long wire.
𝑱⃗
3) Consider a uniform charged rod of charge 𝑄 and length 𝐿, 𝜆ൌ𝑄/𝐿, on
the z‐axis, centered at the origin, extending from 𝑧 ൌ െ
ଶ
to 𝑧 ൌ
ଶ
.
a) Write an integral over the charged rod for the electric potential
on the z‐axis. Calculate the electric potential due to the rod on
the z‐axis for all 𝑧
ଶ
.
b) Expand your result for 𝜙ሺ𝑧ሻ in powers of 𝐿/ሺ2 𝑧ሻ.
It will be useful to know that: (If both expressions don’t help you,
you’ll probably want to redo your integral from part (a).)
lnሺ1 െ 𝑥ሻ ൌ െ 𝑥
𝑛
ஶ
ୀଵ
, lnሺ1 𝑥ሻ ൌ ሺെ1ሻିଵ 𝑥
𝑛
ஶ
ୀଵ
c) Using this expansion, you can determine a sum for the potential 𝜙ሺ𝑟, 𝜃ሻ everywhere.
Note that for points along the 𝑧 axis:
𝜙ሺ𝑧ሻ ൌ 𝜙ሺ𝑟, 𝜃 ൌ 0ሻ
The potential everywhere is given by:
𝜙ሺ𝑟, 𝜃ሻ ൌ 𝑎
𝑟ାଵ 𝑃ሺcos 𝜃ሻ
ஶ
ୀ
Using the result from part (b), determine an expression for all the coefficients 𝑎.
Note: 𝑃ሺcosሺ0ሻሻ ൌ 𝑃ሺ1ሻ ൌ 1 for all 𝑙.
d) Using the multipole expansion for the electric potential, calculate the monopole, dipole, and
quadrupole potential terms due to the line charge. Show that these terms agree with part (c).
4) A square current‐carrying loop with sides of length 𝑙 and counter‐
clockwise current 𝐼 is in the x‐y plane and centered at the origin.
Calculate the magnetic field at the point 𝑟⃗ൌ𝑑 𝑥ො.
a) What is the magnetic field at the point 𝑟⃗ൌ𝑑 𝑥ො if you
approximate the loop by a magnetic dipole?
b) Use cross products to determine the direction of the
magnetic field due to each side of the loop at 𝑟⃗.
c) Write down integrals for the magnetic field at 𝑟⃗ due to
each side of the current loop.
d) Solve for the magnetic field due to the loop at 𝑟⃗ . It might be useful to know that:
න 𝑑𝑥
ሺ𝑎ଶ 𝑥ଶሻ
ଷ
ଶ
ൌ 𝑥
𝑎ଶ √𝑎ଶ 𝑥ଶ
Note: The results are somewhat messy.
e) Expand your result above for 𝑑≫𝑙 ቀ
ௗ ≪ 1ቁ to the lowest non‐zero power in 1/𝑑. Show
that this gives the dipole approximation from part (a).
Hint: It’s a good idea to expand the fields from the right and left wires together, and the top
and bottom wires together.
𝑥
𝑦
𝐼
𝑙
𝑙
𝑥ൌ𝑑
𝑦
𝑧
𝑥
𝜆
