A thin nonconducting spherical shell of radius R1 carries a total charge q1 that is uniformly distributed on its surface. A second, larger thin non-conducting spherical shell of radius R2 that is concentric with the first carries a charge q2 that is uniformly distributed on its surface. Required:
a. Use Gauss's law to find the electric field in the regions r < R1, R1 < r < R2, and r > R2.
b. What should the ratio of the charges q1/q2 and their relative signs be for the electric field to be zero for r > R2?

Explanation:

a )

in the regions r < R₁

charge q inside sphere of radius R₁ = 0

Applying gauss's law for electric field E at distance r <R₁

electric flux through Gaussian surface of radius r = 4π r² E

4π r² E = q / ε₀ = 0 / ε₀

E = 0

Applying gauss's law for electric field E at distance R₁ < r < R₂ .

charge q inside sphere of radius R₁ = q₁

Applying gauss's law for electric field E at distance R₁ < r < R₂

electric flux through Gaussian  surface of radius r = 4π r² E

4π r² E = q₁ / ε₀

E = q₁ / 4πε₀

in the regions r> R₂

charge q inside sphere of radius R₂ = (q₁ + q₂)

Applying gauss's law for electric field E at distance r > R₂

electric flux through Gaussian surface of radius r = 4π r² E

4π r² E = (q₁ + q₂) / ε₀

E =  (q₁ + q₂) /4π ε₀

b )

For electric flux to be zero at r > R₂

(q₁ + q₂) /4π ε₀  = 0

q₁ + q₂ = 0

q₁ / q₂ = - 1 .

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