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
$$\phi_2=\frac{q_1}{4\pi\epsilon_0r_2}=\phi_1\frac{r_1}{r_2}$$