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An infinite plane sheet carrying uniform surface charge density σ slices through a spherical Gaussian surface of radius R, the plane lying a perpendicular distance x from the sphere's centre. What is the total electric flux Φ passing through the spherical surface?

1. pi*R²*σ/ε0
2. 2*pi*(R² - x²)*σ/ε0
3. pi*(R - x)²*σ/ε0
4. pi*(R² - x²)*σ/ε0

Correct answer: pi*(R² - x²)*σ/ε0

Solution

The portion of the charged plane inside the sphere is a disc. The plane is at distance x from the centre, so the circle where it intersects the sphere has radius rho = sqrt(R² - x²). The enclosed charge is q = σ * area of disc = σ * pi*(R² - x²). By Gauss's law, Φ = q/ε0 = pi*(R² - x²)*σ/ε0.

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