# The Electric Potential Outside a Sphere With Given Potential

A conducting spherical shell of radius R is cut in half. The two hemispherical pieces are electrically separated from each other but are left close together, so that the distance separating the two halves may be neglected. The upper half is maintained at Phi=Phi0 and the lower half is maintained at phi=0. Find the electrostatic potential phi at all points in space outside of the surface of the conductors. You may neglect 1/r^4.

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#### Solution Preview

Problem: Given a charged sphere of radius R with electric potential on its surface given by

(1) φ(R, θ) = φ0 if 0 < θ < π/2

0 if π/2 < θ < π,

find φ(r, θ) for all r > R, i.e. everywhere outside the sphere.

Solution: Since the electric potential on the surface of the sphere does not depend on the azimuthal angle ϕ and since φ approaches zero as r goes to infinity, ...

#### Solution Summary

We calculate the electric potential at all points outside a sphere in which the lower half is grounded and the upper half is kept at a fixed potential phi_0. Equations are provided to aid in the understanding of the calculations.