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Preliminary Examination
Preliminary Examination
Preliminary Examination
Preliminary Examination
Preliminary Examination
E&M B2
B2 Suppose the electric field inside a large
piece of isotropic dielectric is 0E , so that the
electric displacement is 0 0 0
D E P . Now
a long, thin, needle‐shaped cavity is hol‐
lowed out of the material. This cavity runs
parallel to P. We assume the polarization is
“frozen in”, so it doesn’t change when the
cavity is made. We also assume the cavity is
small enough that 0 0
, , and P E D are essen‐
tially uniform in the solid.
a. Find the electric field vector E at the center of the cavity in terms of 0E and P.
b. Find the electric displacement vector D at the center of the cavity in terms of 0
D and P.
Part a. The boundary condition for the E field is above below
0
ˆ
E E n , but here bˆ 0 P n ,
so in needle solid 0 E E E .
Part b. We have no polarization in the cavity, so in needleP 0 .
Hence, in needle 0 in needle in needle 0 0 0 D E P E D P
Preliminary Examination
E&M B4
B4 Consider a sphere of radius R with a constant uniform magnetization M. The magnetic
field inside the sphere is given by 203
B M .
a. Calculate the surface current density at any point on the surface.
b. Calculate the tangential component of the B field just outside the sphere.
c. Calculate the normal component of the B field just outside the sphere.
SOLUTION
We’ll need
We also have ˆ ˆ ˆ ˆˆ ˆ ˆ ˆˆ (cos sin ) sin sin ( ) sin z r r θ r θ r φ φ
Define these quantities: (i)
(i)
(o)
(o)
inside field perpendicular
inside field parallel
outside field perpendicular
outside field parallel
B
B
B
B
(i) (i)
(i) 2 2 2 2 20 0 0 0 03 3 3 3 3
ˆ ˆˆ ˆˆ (cos sin ) cos sinM M M M
B B
B M z r θ r θ
b b, b,
NOTE: this is !
b, b,
ˆˆ ˆˆ sin
ˆso sin and
M M
M
K K K M n z r φ
K φ K 0
(o) (i)
(o) (i) 203
is 0
(o) (i)
0
(o) (i) 2 10 0 0 03 3
ˆ ˆ so cos
ˆˆˆ ˆsin sin
ˆ ˆ ˆ ˆsin sin sin sin
M
M M
M M M M
B BK n B B r
B BK n φ r θ
B B θ θ θ θ
