Two metal spheres, one of radius R and the other of radius 2R, both have same surface charge density σ. They are brought in contact and separated. What will be new surface charge densities on them?

Given

Radius of sphere 1 = R

Radius of sphere 2 = 2R

Surface charge density on both spheres = σ

We have two spheres with surface charge equal surface charge density σ, then if the charges on sphere with radius R is Q₁ and charge on sphere with radius 2R is Q₂,

and

Now let the charges on the spheres after getting in contact be Q₁’ and Q₂’ then from the conservation of charge, we have

Since the spheres initially have unequal potentials, when they come in contact charges flow from higher potential to lower potential until there potentials are equal, thus for equal potential,

Substituting the relationship between and in (1), we get

and

Therefore, the new surface charge densities are

and