How can I solve this please? Help Needed!

Question

An isolated charged conducting sphere has a radius R = 16.0 cm. At a distance of r = 22.0 cm from the center of the sphere the electric field due to the sphere has a magnitude of E = 4.90 ✕ 10⁴ N/C.

A) What is its surface charge density (in µC/m²)?

_____ µC/m²

B) What is its capacitance (in pF)?

_____ pF

C) What If? A larger sphere of radius 25.0 cm is now added so as to be concentric with the first sphere. What is the capacitance (in pF) of the two-sphere system?

_____ pF

Answer 1

a. First let's get the charge on the sphere

As E = kQ/r², Q = Er²/k = 4.90 x 10⁴ x 0.22² / 8.99 x 10⁹ = 2.64 x 10^-7 C. This charge can be considered to be at the center of the sphere.

One way to get the charge density is to divide the charge by the area of the sphere.

σ = 2.64x10^-7 / 4πx0.16² = 8.20 x 10^-7 C/m². I also used σ = ε^o x E and got the same result.

b. The capacitance of the sphere can be obtained from

C = 4πε^o / r^-1 = 12.56 x 8.85 x 10-12 /0.16^-1 = 1.78 x 10^-11 F or 17.8 pF.

c. The equation for a spherical capacitor is C = 4πε^o / (1/0.16 - 1/0.25) x 10¹² = 49.4 pF

check math.