Question

Two long, charged, thin-walled, concentric cylindrical shells have radii of 3.9 and 9.4 cm. The charge per unit length is 6.8 × 10^-6 C/m on the inner shell and -8.5 × 10^-6 C/m on the outer shell. What are the (a) magnitude E and (b) direction (radially inward or outward) of the electric field at radial distance r = 5.9 cm? What are (c) E and (d) the direction at r = 14 cm?

Answer 1

Given,

Inner radius, R₁ = 3.9 cm = 0.039 m

Outer radius, R₂ = 9.4 cm = 0.094 m

Charge per unit length of inner cylinder, ₁ = 6.8 * 10^-6 C/m

Charge per unit length of outer cylinder, ₂ = -8.5 * 10^-6 C/m

a)

Given

r = 5.9 cm = 0.059 m

Since, r > R₁ but r < R₂

Hence, Electric field at r = 5.9 cm will be due to inner cylindercal shell, and electric field due to outer cylinderical shell is zero.

Electric field at any distance r due to long cylinderical shell is given by

                               E = / 2₀r

Thus, Electric field due to inner shell, E₁ = ₁ / 2₀r

                                                                  = ( 6.8 * 10^-6) / ( 2* 3.14 * (8.85 * 10^-12) * 0.059 )

                                                                  = ( 6.8 * 10^-6) / ( 3.28 * 10^-12 )

                                                                  = 2.073 * 10⁶ V/m

Thus, magnitude of electric field at r = 5.9 cm is 2.073 * 10⁶ V/m

b)

Since Electric field is positive, hence the direction is radially outwards.

c)

Now,

r = 14 cm = 0.14 m

Since,

r > R₁ and r > R₂

Hence, electric field at r = 14 cm is due to both the shells.

As we know,

Electric field at any distance r due to long cylinderical shell is given by

                               E = / 2₀r

Electric field at r = 14 cm,

                                E₂ = electric field due to inner shell + electric field due to outer shell

=> E₂ =( ₁ / 2₀r ) + ( ₂ / 2₀r ) = ( ₁ + ₂ )/ 2₀r

           = ( 6.8 * 10^-6 + ( -8.5 * 10^-6)) / ( 2 * 3.14 * (8.85 * 10^-12) * 0.14 )

           = ( -1.7 * 10^-6 ) / ( 7.78 * 10^-12 )

           = - 0.2185 * 10⁶ = -2.185 * 10⁵ V/m

Thus, magnitude of electric field at r = 14 cm is 2.185 * 10⁵ V/m

d)

Since, value of electric field at r = 14 cm is negative

Hence, the direction of electric field is radially inwards.