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Electricity and Magnetism I (P331)
M. R. Shepherd
October 17, 2008
A capacitor with a nonuniform dielectric
Consider a parallelplate capacitor ﬁlled with nonuniform dielectric. The
dielectric is still linear, in other words,
P
=
±
0
χ
e
E
, but this time
χ
e
varies
as a function of position in the dielectric. Let’s say that
χ
e
varies such that
±
r
=
±
r
0
+
ax.
Here
±
r
0
is a dimensionless constant,
x
is the distance from one plate, and
a
is constant with units 1
/
length. Let’s assume the area of the plates is
A
and their separation is
d
. What is the capacitance?
x=0
x=d
x
Remember the strategy for computing capacitance: put a charge
Q
on the capacitor and then compute
the potential diﬀerence between the plates in terms of this charge. The capacitance can then be identiﬁed
by comparing this expression to
V
=
Q/C
. In order to compute
V
we ﬁrst need to ﬁnd the electric ﬁeld.
Let’s charge up the plates so that we have a free surface charge density of +
σ
f
on the left plate and

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