An insulating spherical shell has an inner radius R1 and an outer radius R2. The shell has a uniform charge volume density of 10.0 nC/m3 and the electric field just outside the shell’s outer surface is radially directed and has a uniform strength of 120 N/C. A. If the shell’s outer radius is R2 = 1.1 m, what is its inner radius R1, in meters?

0.782m

Explanation:

Parameters given:

Volume charge density, ρ = 10 * 10^(-9) C/m³

Electric field outside the shell, E = 120 N/C

Outer radius, R2 = 1.1m

Inner radius, R1 =?

Volume charge density, ρ is given as:

ρ = Q/V

V = volume of shell = 4/3 * π * (R2 - R1)³

Therefore,

ρ = Q/[4/3 * π * (R2 - R1)³] (1)

The electric field outside the shell is given as:

E = kQ/(R2 - R1)² (2)

We can make charge, Q, the subject of the formula:

From (1):

Q = 4ρ/3 * π * (R2 - R1)³

From (2):

Q = E/k * (R2 - R1)²

Equating both formulae:

4ρ/3 * π * (R2 - R1)³ = E/k * (R2 - R1)²

=> (R2 - R1)³/(R2 - R1)² = 3E/(4πρk)

(R2 - R1) = 3E/(4πρk)

=> R1 = R2 - 3E/(4πρk)

Inserting the values of all parameters, k = Coulumbs constant:

R1 = 1.1 - (3 * 120)/(4 * π * 10 * 10^(-9) * 9 * 10^9)

R1 = 1.1 - 360/1131.12

R1 = 1.1 - 0.318

R1 = 0.782m

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