P HYSICS 301: I NTERMEDIATE E LECTROMAGNETISM
1. Short Answer – 30 points total
FALL 2010, A SSESSMENT #3 (F INAL )
None of the problems in this section should require an involved calculation or a complicated integral.
If you find yourself attempting one, see if you can find a simpler solution. Also beware of extraneous
information!
Pledge and sign:
(a) For most materials, the current density is proportional to the electromagnetic force on the free
!.
charges: J!f = σ f!. Briefly justify the common approximation (Ohm’s Law) that J!f = σ E
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(b) Examine the diagram below. Is this material paramagnetic or diamagnetic, and how do you know?
1. Short Answer – 30 points total
2. Magnetic Field for Coax Cable – 24 points
3. Mutual Inductance – 20 points
4. A Dielectric Bovine – 20 points
5. Faraday’s Law – 24 points
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(Short Answer, continued – 30 points total). None of the problems in this section should require
an involved calculation or a complicated integral. If you find yourself attempting one, see if you
can find a simpler solution. Also beware of extraneous information!
(Short Answer, continued – 30 points total). None of the problems in this section should require
an involved calculation or a complicated integral. If you find yourself attempting one, see if you
can find a simpler solution. Also beware of extraneous information!
(c) We have considered Maxwell’s equations for the electric field in their differential form. Demonstrate
your knowledge of vector calculus to rewrite the differential equations in integral form, for sources
ρ and J!.
! = !0, and any net charge resides on the
(e) In a perfect conductor, the conductivity is infinite, so E
surface (just as it does for an imperfect conductor in electrostatics). Using Maxwell’s equations,
show that the magnetic field is constant in time inside a perfect conductor.
!
! =?
! · da
E
!
! =?
! · dl
E
!
(f) In a superconductor, the conductivity is infinite with the additional property that the (constant) B
inside is in fact zero. Using Maxwell’s equations, find the volume current density in a superconductor. Where is the current flowing?
! (t) = P0 t ẑ and magne(d) A neutral sphere of insulating material has time-dependent polarization P
! = M0 ts2 ẑ , with no free current. Find the magnitude and direction of the total volume
tization M
! in the sphere, using cylindrical coordinates.
current density J(t)
2. Magnetic Field for Coax Cable – 24 points
A long coaxial cable consists of an inner solid conducting cylinder (radius a) and an outer conducting
cylindrical shell (radius b).
Free current I flows uniformly through the inner conductor in the +ẑ direction and back along the outer
cylinder in the -ẑ direction, as shown below. The space between the two cylinders contains a linear
insulating material with permeability µ. The inner and outer conductors have no magnetic susceptibility
(permeability of the conductors is µ0 ).
! everywhere, (ii) the magnetic field B
! everywhere, and (iii) the magnitude and direction of all
Find (i) H
the bound currents, clearly indicating where they are located.
3. Mutual Inductance – 20 points
Two tiny loops of wire each have radius a and carry current I . Treat these current loops as magnetic
dipoles. One is at the origin, with magnetic moment m
! 1 = m1 ẑ , and the other is positioned at z = d,
with magnetic moment m
! 2 = m2 (cos θ ẑ + sin θ ŷ).
Recall that the dipole vector potential and field can be written in spherical coordinates as
! dip (!r) = µ0 m sin θ φ̂,
A
4π r2
! dip (!r) = µ0 m (2 cos θ r̂ + sin θ θ̂).
B
4πr3
Estimate the mutual inductance of the two loops.
4. A Dielectric Bovine – 20 points
A un-charged (neutral) cow is wandering in free space. We model the cow as a sphere, and we neglect
earth, gravity, air resistance, etc. The cow has outer radius b and a centrally-located vacuous empty
stomach of radius a. For a < r < b, the cow is a linear dielectric material with permittivity '.
(a) The spherical cow is a distance s away from an infinite straight electric fence, with line charge
density λ. In a concise calculation using Gauss’ Law, find the flux of the total electric field through
the cow.
(b) The hungry cow approaches a charged daisy, which we will model as a point charge +q . If the
cow started at r = ∞ and stops at a distance of R from the daisy, is there any electrostatic
potential energy stored in the cow-daisy system? Briefly justify your yes/no answer with words
and/or math.
(c) The cow has eaten the point-charge daisy, which now sits at the very center of the cow’s stomach.
! everywhere and find the total electric field E
! for each region: (i) r < a inside the otherwiseFind D
empty stomach cavity, (ii) a < r < b in the cow, and (iii) r > b outside the cow.
(d) Find the electrostatic potential in the region a < r < b. What reference point do you use?
5. Faraday’s Law – 24 points
A square loop of wire (side a, total resistance R) is initially a distance s0 away from a long neutral wire
carrying current I! = I0 ẑ , as shown.
(a) Find the flux of the magnetic field through the loop.
(b) The loop is pulled directly away from the wire, with constant velocity !v = v0 ŝ.
Find (i) the electromotive force generated by the change in flux; and (ii) the magnitude and direction of the current that flows in the loop.
(c) The loop is pulled parallel to the wire, with constant velocity !v = v0 ẑ .
What is the electromotive force?
(d) The current in the long wire now increases slowly in the z direction: I! = I0 (1 − e−t/τ )ẑ .
Explain which direction the loop could be moved to prevent current from flowing in the loop.