Question

A small spherical shell S₁ has point charges q₁ = −3 μC, q₂ = −2 μC and q₃ = 9 μC inside it. This shell is enclosed by another big spherical shell S₂. A point charge Q is placed in between the two surfaces S₁ and S₂. If the electric flux through the surface S₂ is four times the flux through surface S₁, find charge Q.

Answer 1

Given, charges inside the small spherical shell S₁:

q₁ = −3 μC, q₂ = −2 μC and q₃ = 9 μC

Charge Q is placed between the two shells.

The electric flux through S₂ is four times the flux through S₁.

According to Gauss’s law:

Φ = `Q_"enc"/epsilon_0`

The total charge enclosed by S₁:

Q_enc, S₁ = (−3 + (−2) + 9)μC

= −5 + 9 μC

= 4 μC

`Phi_(S_1) = (4 mu C)/epsilon_0`

Flux through S₂:

`Phi_(S_2) = 4 Phi_(S_1)`

Substituting `Phi_(S_1)`:

Q_enc, `S_2/epsilon_0 = 4 xx (4 mu C)/epsilon_0`

Q_enc, S₂ = 16 μC

Since the charge enclosed by S₂ includes both the charge inside S₁ and the additional charge Q:

Q_enc, S₁ = Q_enc, S₁ + Q

16 μC = 4 μC + Q

Q = 16 μC − 4 μC

Q = 12 μC

So, the charge Q must be 12 μC to satisfy the given flux condition.