Question

FIGURE EX24.27 shows a hollow cavity within a neutral conductor. A point charge $Q$is inside the cavity. What is the net electric flux through the closed surface that surrounds the conductor?

Short answer

The net electric flux through the closed surface is${\Phi }_e=\boxed{\frac{\mathbf{Q}}{{\mathbf{\epsilon }}_{\mathbf{0}}}}$

Step-by-step solution

Given information and Theory used 

Given figure :

Theory used :

The amount of electric field that travels through a closed surface is referred to as the electric flux. The electric flux through a surface is proportional to the charge inside the surface, according to Gauss's law, which is provided by equation :

${\Phi }_e=\oint \overrightarrow{E}·d\overrightarrow{A}=\frac{Q_{in}}{{\epsilon }_0}$ (1)

Calculating the net electric flux through the closed surface 

The cavity's positive charge $Q$induces a negative charge$-Q$on the conductor's inner side.

As seen in the diagram below, the induced negative charge produces a positive charge $Q$at the conductor's surface.

$Q_{in}=Q+(-Q)+Q=Q$is the total enclosed charge through the closed surface that surrounds the conductor.

To calculate the electric flux, enter this number into equation (1).

${\Phi }_e=\frac{Q_{in}}{{\epsilon }_0}=\boxed{\frac{\mathbf{Q}}{{\mathbf{\epsilon }}_{\mathbf{0}}}}$