A solid conducting sphere with radius R that carries positive charge Q is concentric with a very thin insulating shell of radius 2R that also carries charge Q. The charge Q is distributed uniformly over the insulating shell. Find the magnitude of the electric field in the region 02R. Express your answer in terms of the variables R, r, Q, and constants π and ε0.

Gauss Law relates the distribution of electric charge to the resulting electric field.

Applying Gauss's Law,

EA = Q / ε₀

Where:

E is the magnitude of the electric field,

A is the cross-sectional area of the conducting sphere,

Q is the positive charge

ε₀ is the permittivity

We be considering cases for the specified regions.

Case 1: When r < R

The electric field is zero, since the enclosed charge is equal to zero

E (r) = 0

Case 2: When R < r < 2R

The enclosed charge equals to Q, then the electric field equals;

E (4πr²) = Q / ε₀

E = Q / 4πε₀r²

E = KQ / r²

Constant K = 1 / 4πε₀ = 9.0 * 10⁹ Nm²/C²

Case 3: When r > 2R

The enclosed charge equals to Q, then the electric field equals;

E (4πr²) = 2Q / ε₀

E = 2Q / 4πε₀r²

E = 2KQ / r²