Question

A cylindrical capacitor consists of a solid inner conducting core with radius 0.210 cm , surrounded by an outer hollow conducting tube. The two conductors are separated by air, and the length of the cylinder is 10.0 cm . The capacitance is 37.0 pF .

A) Calculate the outer radius of the hollow tube.

B)When the capacitor is charged to 140 V , what is the charge per unit length λ on the capacitor?

Answer 1

(A) The outer radius of the hollow tube which is given as :

using an equation, C = 2 K ₀ L / ln (R / r)                                        { eq.1 }

where, C = capacitance = 37 x 10^-12 F

K = dielectric constant = 1 (for vaccum)

₀ = permittivity for vaccum = 8.85 x 10^-12 C²/Nm²

L = length of the cylinder = 10 cm = 0.1 m

r = inner radius of the cylinder = 0.21 cm

inserting all these values in above eq.

(37 x 10^-12 F) = 2 (3.14) (1) (8.85 x 10^-12 C²/Nm²) (0.1 m) / ln [R / (0.21 cm)]

ln [R / (0.21 cm)] = (5.55 x 10^-12 C²/Nm) / (37 x 10^-12 F)

ln [R / (0.21 cm)] = 0.15

[R / (0.21 cm)] = 1.161

R = (1.161) x (0.21 cm)

R = 0.243 cm

(B) When the capacitor is charged to 140 V , then the charge per unit length on the capacitor will be given as :

charge enclosed by the surface, Q = λ L                                                   { eq.2 }

Q / L = λ

where, Q / L = charge per unit length

λ = C V / L                                                                                                 { eq.3 }

inserting the values in eq.3,

λ = (37 x 10^-12 F) (140 V) / (0.1 m)

λ = 51800 x 10^-12 F

λ = 5.18 x 10^-8 F