Question

If \(∮_s\overrightarrow{E}.\overrightarrow{dS}=0\) over a surface, then:

• A. the electric field inside the surface is necessarily uniform
• B. the number of flux lines entering the surface must be equal to the number of flux lines leaving it
• C. the magnitude of electric field on the surface is constant
• D. all the charges must necessarily be inside the surface

Answer 1

The Correct Option is B

To solve the problem, we need to understand the context of the given expression:

\(∮_s\overrightarrow{E}.\overrightarrow{dS}=0\)

This expression is a mathematical statement of Gauss's Law for Electric Fields in integral form, where \(∮_s\) denotes the closed surface integral of the electric field \(\overrightarrow{E}\) over the surface \(S\). The expression equates to zero, indicating that the net electric flux through the surface is zero.

1. Understanding Gauss's Law: Gauss's law relates the electric flux through a closed surface to the charge enclosed by that surface. Mathematically, it is given by:
   \(∮_s\overrightarrow{E}.\overrightarrow{dS} = \frac{Q_{\text{enclosed}}}{ε_0}\)
2. Interpreting the Given Condition: If the net electric flux through a surface is zero, i.e., \(∮_s\overrightarrow{E}.\overrightarrow{dS}=0\), this implies that the net charge \(Q_{\text{enclosed}}\) within the surface is zero.
3. Implication for Electric Field Lines: Since the net charge inside is zero, the number of electric field lines entering the surface must be equal to the number of field lines leaving it.
4. Analyzing the Options:
   • The electric field inside the surface is necessarily uniform: This is not necessarily true. The electric field can be non-uniform, but the net flux being zero depends on the net charge being zero.
   • The number of flux lines entering the surface must be equal to the number of flux lines leaving it: Correct. As explained above, this implies net zero charge within.
   • The magnitude of the electric field on the surface is constant: This is not true, as the electric field could vary over the surface.
   • All the charges must necessarily be inside the surface: Not true. There can be no charge inside as well, leading to net zero flux.

Thus, the correct answer is that the number of flux lines entering the surface must be equal to the number of flux lines leaving it.