Shown is a coaxial cable, made of an inner cylindrical conducting wire of radius

Question

Shown is a coaxial cable, made of an inner
cylindrical conducting wire of radius a and
an outer conducting sheath of inner radius
b. The wire carries uniform positive charge per unit length ? . The

sheath is grounded (V = 0). Also shown is a cross section, with a point (indicated by the dot) at distance r from the symmetry axis.

a. What is the charge per unit length on the inner surface of the
sheath? What is the direction of E at the indicated point? How do you know?

b. Use Gauss’s law (using a cylindrical surface passing through the point) to find the magnitude of the E-field at the point indicated. Ans: E = 2k? / r .

c. Find the potential on the wire’s surface. Ans: V(a) = 2k? ? ln(b/ a) .

Shown is a coaxial cable, made of an inner cylindrical conducting wire of radius a and an outer conducting sheath of inner radius I. The wire carries uniform positive charge per unit length A. The sheath is grounded (V 0). Also shown is a cross section, with a point (indicated by the dot) at distance r from the symmetry axis. What is the charge per unit length on the inner surface of the sheath? What is the direction of E at the indicated point? How do you know? Use Gauss's law (using a cylindrical surface passing through the point) to find the magnitude of the E-field at the point indicated. Ans: E-2kX/r b. Find the potential on the wire's surface. Ans: V(a)-2kX In(b / a) C.

Explanation / Answer

a]  charge per unit length on the inner surface = - lambda.

By application of Gauss law, in order to make electric field zero in outer conducting sheath, inner charge density should be -lambda.

Direction of E is radially outward as charge lambda is positive.

b] By taking cylinderical gaussian surface having concentric radius r,

E*area = Qenclosed/e0

E* 2pi r L = lambda*L/e0

E = lambda/ 2pi r e0

= 2k lambda/r

c] V = integral -Edr

= 2k lambda ln r

Potential difference between wire and sheath = 2k lambda ln(b/a)

Since V at sheath is zero, V(a) = 2k lambda ln(b/a)

  

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