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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?
Solution
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
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