An infinitely long cylinder in free space is concentric with the z-axis and has radius a. The net charge density p in this cylinder is given in cylindrical coordinates by, 1 a² + ² where A is a constant. (a) Show that the total charge per unit length, À in the cylinder is p(r) = A for r
An infinitely long cylinder in free space is concentric with the z-axis and has radius a. The net charge density p in this cylinder is given in cylindrical coordinates by, 1 a² + ² where A is a constant. (a) Show that the total charge per unit length, À in the cylinder is p(r) = A for r
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Transcribed Image Text:An infinitely long cylinder in free space is concentric with the z-axis
and has radius a. The net charge density p in this cylinder is given in
cylindrical coordinates by,
1
a² +r²
where A is a constant.
(a) Show that the total charge per unit length, λ in the cylinder is
λ = πA ln 2.
p(r) = A-
Hint: you may find the following integral useful.
1
2
J
for r <a,
X
a² +x²
-dx In(a² + x²) + constant.
(b) Explicitly using symmetry principles, show that the electric field
due to this charge distribution has the form
E(r) = Er(r)er.
(c) Determine Er(r) outside the cylinder (r > a) and inside the
cylinder (r< a).
(d) The cylinder is composed of a material in which the polarisation P
is given by
P = P₁² in (1 +5²) e₁₁
er,
r
where Po is a constant.
Determine the bound charge density pb in the cylinder. Hence, or
otherwise, determine a relation between A and Po such that the
free charge density of in the cylinder vanishes.
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