Question

An infiinitely long solid conducting cylindrical shell of radius a = 2.2 cm and negligible thickness is positioned with its symmetry axis along the z-axis as shown. The shell is charged, having a linear charge density λ_inner = -0.34 μC/m. Concentric with the shell is another cylindrical conducting shell of inner radius b = 13.8 cm, and outer radius c = 15.8 cm. This conducting shell has a linear charge density λ _outer = 0.34μC/m.

1) What is E_x(P), the x-component of the electric field at point P, located a distance d = 6.8 cm from the origin along the x-axis as shown?

2) What is V(c) – V(a), the potential difference between the the two cylindrical shells?

3) What is C, the capacitance of a one meter length of this system of conductors?

4) The magnitudes of the charge densities on the inner and outer shells are now changed (keeping λ_inner = -λ_outer) so that the resulting potential difference doubles (V_ca,new = 2V_ca,initial). How does C_new, the capacitance of a one meter length of the system of conductors when the charge density is changed, compare to C, the initial capacitance of a one meter length of the system of conductors?

C_new < C

C_new = C

C_new > C

5) What is λ_outer,new ?

Answer 1

1) for a < r < b

E (2 pi r L ) = Qin / (e0)

E( 2 pi 0.068 L ) = (0.34 x 10^-6 x L ) / (8.854 x 10^-12)

E = 89877.4 N/C towards the common axis.

Ex = - 89877.4 N/C

(2) V(c) - V(a) = V(b) - V(a)

becuase V(c) = V(b)

and deltaV = - integral of E.dr

E = ( lambda ) / (2 pi e0 r ) =

V = (lambda / 2 pi e0) ln(b/a)

= (0.34 x 10^-6 / (2 xpi x 8.854 x 10^-12) ) ln(0.138/0.02)

= 11222.3 Volt

(3) Q = C V

lambda L = C (11222.3)

C / L = 3.03 x 10^-11 F/m

(4) now now V' = 2 V
and V = (lambda / 2 pi e0) ln(b/a)
hecne lambda is doubled.

but C will be constant.

{ C is independent of charge densities}

(5) Youter = 2 x 0.34 uC / m = 0.68 uC/m