Problem 85 - Master Resource Book in JEE Main Physics

Question

\(\oint \mathbf{E} \cdot d \mathbf{S}=0\) over a surface, then \(\quad\) [NCERT Exemplar] (a) the electric field inside the surface and on it is zero (b) the electric field inside the surface is necessarily uniform (c) the number of flux lines entering the surface must be equal to the number of lux lines leaving it (d) all charges must necessarily be outside the surface

Step-by-Step Solution

Answer

(c) The number of flux lines entering the surface must be equal to the number of flux lines leaving it.

Step 1: Understanding the Integral

The given expression \( \oint \mathbf{E} \cdot d \mathbf{S} = 0 \) is an application of Gauss's law for electric flux, which relates the electric field \( \mathbf{E} \) passing through a closed surface to the charge enclosed by that surface.

Step 2: Applying Gauss's Law

According to Gauss's Law, the electric flux \( \Phi \) through a closed surface is given by \( \Phi = \oint \mathbf{E} \cdot d \mathbf{S} = \frac{Q_{enc}}{\varepsilon_0} \), where \( Q_{enc} \) is the total charge enclosed by the surface and \( \varepsilon_0 \) is the permittivity of free space.

Step 3: Interpreting Zero Flux

Since the electric flux \( \Phi = 0 \) over the closed surface, it implies that the net charge \( Q_{enc} \) inside the surface must be zero. There could be positive and negative charges, but their net sum must be zero inside the surface.

Step 4: Conclusion with Multiple-Choice

Given the condition that the net charge inside the surface is zero, the appropriate conclusion is that the number of electric field lines entering the surface equals the number of lines exiting the surface, making the correct answer (c) because it indicates zero net flux, consistent with no net charge inside.

Key Concepts

Electric Flux

Electric flux represents how many electric field lines pass through a given surface. It is a measure of the number of electric field lines transversing the surface. To calculate electric flux, you would use the formula:\[ \Phi = \oint \mathbf{E} \cdot d \mathbf{S} \]where \( \Phi \) is the electric flux, \( \mathbf{E} \) is the electric field, and \( d \mathbf{S} \) is a small area vector on the surface.

This formula shows that the electric flux depends on both the strength of the electric field and the size and orientation of the surface it is passing through. If the surface is oriented perpendicularly to the electric field lines, you'll have maximum flux. Conversely, if it's parallel, there is no flux passing through it. So, understanding electric flux helps visualize how electric fields interact with different surfaces.

Closed Surface

A closed surface is like a balloon that completely encloses a certain space. It has no openings or edges, forming a finite boundary. In terms of Gauss's Law, this means the analysis involves an entire 3D surface, rather than any part of it.

For example, consider a sphere or a cube. These can be imagined as closed surfaces. The importance of a closed surface in Gauss's Law is that it helps to apply the law to determine the net charge within this boundary.

• All electric flux entering must also leave, assuming no charge inside.
• If there is any charge within, it affects the net flux through the surface.

Understanding the concept of a closed surface will guide you in identifying the boundaries for various problems, simplifying calculations and conceptual visualization.

Net Charge

Net charge refers to the total amount of electric charge within a closed surface. It is a critical element in Gauss's Law because it directly influences the electric flux.

According to Gauss's Law, the net charge \( Q_{enc} \) enclosed by a surface is related to the electric flux \( \Phi \) through:\[ \Phi = \frac{Q_{enc}}{\varepsilon_0} \]where \( \varepsilon_0 \) is the permittivity of free space.

• If the net charge within the surface is zero, this implies no net electric flux, meaning as much flux enters as leaves.
• A positive net charge would result in more flux leaving the surface than entering.
• A negative net charge implies more flux entering than leaving.

Understanding net charge helps in analyzing the effect of electric fields on the enclosed region.

Electric Field Lines

Electric field lines are imaginary lines that represent the direction and strength of the electric field. The density of these lines indicates how strong the electric field is in that region.

Here are some basic rules about electric field lines:

• They originate from positive charges and terminate at negative charges.
• The closer the lines, the stronger the electric field.
• They never cross each other.

In relation to Gauss's Law, the behavior of these field lines with respect to a closed surface provides insight into the net charge within. For a surface with zero net charge inside, the number of lines entering equals the number of lines exiting. This is why field lines are an essential visual tool to understand electric fields and their effects on surroundings.