Question

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?

Answer 1

The charges on metal spheres before contact, are `Q_1 = σ * 4piR^2`

And `Q_2 = σ * 4pi(2R)^2 = 4(σ * 4piR^2) = 4Q_1`

Let the charges on the metal spheres, after coming in contact becomes `Q"'"_1` and `Q"'"_2`

Applying law of conservation of charges,

`Q"'"_1` + `Q"'"_2 = Q"'"_1` + `Q"'"_2 = 5Q_1 = 5(σ * 4piR^2)`  ......(i)

When metal spheres come in contact, they acquire equal potentials. Therefore, we have

`1/(4piε_0) (Q"'"_1)/R = 1/(4piε_0) (Q"'"_2)/(2R)` ⇒ `Q"'"_1 = (Q"'"_2)/2`   ......(ii)

On solving (i) and (ii), we get

∴ `Q"'"_1 = 5/3 (σ * 4piR^2)` and `Q"'"_2 = 10/3 (σ * 4piR^2)`

∴ `σ_1 = (5σ)/3` and `σ_2 = (5)/3` σ