Potential of Metal Sphere

A metal sphere having a radius \(r_1\) charged to a potential \(\phi_1\) is enveloped by a thin-walled conducting spherical shell of radius \(r_2\). Determine the potential \(\phi_2\) acquired by the sphere after it has been connected for a short time to the shell by a conductor.

Solution:

The charge \(q_1\) of the sphere can be determined from the relation $$ q_1=4\pi\epsilon_0r_1$$
After the connection of the sphere to the envelope, the entire charge \(q_1\) will flow from the sphere to the envelope and will be distributed uniformly over its surface.
Its potential \(\phi_2\) (coinciding with the new value of the potential of the sphere) will be
$$ \phi_2=\frac{q_1}{4\pi\epsilon_0r_2}=\phi_1\frac{r_1}{r_2}$$

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