Download Homework No. 07 (Spring 2015) PHYS 420: Electricity and Magnetism II

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Transcript
Homework No. 07 (Spring 2015)
PHYS 420: Electricity and Magnetism II
Due date: Monday, 2015 Apr 6, 4.30pm
1. (40 points.) (Based on Problem 5.58, Griffiths 4th edition.) A circular loop of wire
carries a charge q. It rotates with angular velocity ω about its axis, say z-axis.
(a) Show that the current density generated by this motion is given by
q
J(r) =
ω × r δ(ρ − a)δ(z − 0).
2πa
(1)
Hint: Use J(r) = ρ(r)v, and v = ω × r for circular motion.
(b) Using
Z
1
m=
d3 r r × J(r).
2
determine the magnetic dipole moment of this loop to be
m=
qa2
ω.
2
(2)
(3)
(c) Calculate the angular momentum of the rotating loop to be
L = ma2 ω,
(4)
where m is the mass of the loop.
(d) What is the gyromagnetic ratio g of the rotating loop, which is defined by the relation
m = gL.
2. (20 points.) A charged spherical shell carries a charge q. It rotates with angular velocity
ω about a diameter, say z-axis.
(a) Show that the current density generated by this motion is given by
q
ω × r δ(r − a).
J(r) =
4πa2
(5)
Hint: Use J(r) = ρ(r)v and v = ω × r for circular motion.
(b) Using
Z
1
d3 r r × J(r).
m=
2
determine the magnetic dipole moment of the rotating sphere to be
m=
1
qa2
ω.
3
(6)
(7)
3. (30 points.) A steady current I flows down a long cylindrical wire of radius a. The
current density in the wire is described by, n > 0,
J(r) = ẑ
ρ n
I
(n
+
2)
θ(a − ρ).
2πa2
a
(8)
(a) Show that, indeed,
Z
dS · J(r) = I.
(9)
S
(b) Using Ampere’s law show that the magnetic field inside and outside the cylinder is
given by

µ0 2I ρ n+2


φ̂ ρ < a,

 4π ρ a
(10)
B(r) =

µ0 2I


φ̂
ρ > a.

4π ρ
(c) Plot the magnitude of the magnetic field as a function of ρ.
2