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

The given integral equation is \(∮_s\overrightarrow{E}.\overrightarrow{dS}=0\), which is an expression of Gauss's Law for electricity in integral form for electric flux through a closed surface. This equation indicates that the net electric flux (total number of electric field lines) through the closed surface is zero.

Let's evaluate the options: 

The electric field inside the surface is necessarily uniform:
This is incorrect. A zero net electric flux through a surface doesn't imply uniformity of the electric field within that surface. The field can vary in different regions, maintaining an overall balance such that net flux remains zero.

The number of flux lines entering the surface must be equal to the number of flux lines leaving it:
Correct. A zero net electric flux indicates that whatever flux enters the surface is equally leaving it, ensuring the total entering equals the total leaving.

The magnitude of electric field on the surface is constant:
This is incorrect. The net flux equating to zero doesn't necessarily imply that the electric field's magnitude remains constant over the surface.

All the charges must necessarily be inside the surface:
This is incorrect. Having zero net electric flux doesn't require charges to be inside the surface. Charges can be outside, affecting the field in a way that flux entering equals flux leaving.

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

Answer 2

The correct option is (B): the number of flux lines entering the surface must be equal to the number of flux lines leaving it

\(\oint\vec{E}.\vec{d}s\) represents electric flux over the closed surface.

When \(\oint\vec{E}.\vec{d}s\)=0, it means the number of flux lines entering the surface, will be equal to the number of flux lines leaving it.

As, \(\oint\vec{E}.\vec{d}s=\frac{q}{\epsilon_0}\) where q is the charge enclosed by the surface.

When \(\oint\vec{E}.\vec{d}s=0\),q=0 i.e, net charge enclosed by the surface must be zero.