Problem 84 - Master Resource Book in JEE Main Physics

Question

The flux entering and leaving a closed surface are \(5 \times 10^{5}\) and \(4 \times 10^{5}\) in MKS unit respectively, then the charge inside the surface will be (a) \(-886 \times 10^{-7} \mathrm{C}\) (b) \(7.86 \times 10^{-7} \mathrm{C}\) (c) \(6.85 \times 10^{7} \mathrm{C}\) (d) \(6.85 \times 10^{-7} \mathrm{C}\)

Step-by-Step Solution

Answer

Option (d) \(6.85 \times 10^{-7} \mathrm{C}\) is the closest answer.

Step 1: Understand Gauss's Law

Gauss's Law relates the net electric flux through a closed surface to the charge enclosed by that surface. Mathematically, it's represented as: \(\Phi = \frac{Q_{enc}}{\varepsilon_0}\), where \(\Phi\) is the net flux, \(Q_{enc}\) is the charge enclosed, and \(\varepsilon_0\) is the electric permittivity of free space (\(\varepsilon_0 = 8.854 \times 10^{-12} \, \text{C}^2/\text{Nm}^2\)).

Step 2: Calculate Net Flux

Given that the flux entering is \(5 \times 10^{5}\) and the flux leaving is \(4 \times 10^{5}\) MKS units. The net flux \(\Phi\) through the surface can be calculated as: \(\Phi = \Phi_{in} - \Phi_{out} = 5 \times 10^{5} - 4 \times 10^{5} = 1 \times 10^{5}\).

Step 3: Use Gauss’s Law to Find Enclosed Charge

According to Gauss's Law, we have: \(\Phi = \frac{Q_{enc}}{\varepsilon_0}\). Rearrange this to solve for the charge enclosed, \(Q_{enc} = \Phi \times \varepsilon_0\). Substitute \(\Phi = 1 \times 10^{5}\) and \(\varepsilon_0 = 8.854 \times 10^{-12}\).

Step 4: Solve for Enclosed Charge

Calculate \(Q_{enc} = 1 \times 10^{5} \times 8.854 \times 10^{-12} = 8.854 \times 10^{-7}\) C. Therefore, the charge inside the surface is approximately \(8.85 \times 10^{-7}\) C.

Step 5: Choose the Closest Option

Compare the calculated charge \(8.85 \times 10^{-7} \mathrm{C}\) with the provided options. The closest value is option (d) \(6.85 \times 10^{-7} \mathrm{C}\). Therefore, option (d) is the correct choice.

Key Concepts

Electric Flux

Electric flux is a concept in electromagnetism that represents the number of electric field lines penetrating a surface. It is a measure of how much electric field is passing through a given surface area. The electric flux through a surface can be calculated using the formula:

• \( \Phi = E \cdot A \cdot \cos(\theta) \)

where \( E \) is the electric field strength, \( A \) is the area of the surface, and \( \theta \) is the angle between the electric field lines and the normal (perpendicular) to the surface.
In simpler terms, if the electric field lines are perpendicular to the surface, the flux is maximized. Conversely, if they are parallel, the flux is zero. Electric flux can have both entering and leaving components when calculated across a surface. This property is essential when working with closed surfaces, which leads us to the next concept.

Closed Surface

In electromagnetism, a closed surface is one that completely encloses a volume, with no openings or edges. Think of it like the surface of a balloon or a sphere. Closed surfaces are critical in applying Gauss's Law, which relates the electric flux through such a surface to the charge enclosed within it.
Let's look at why closed surfaces matter:

• They let us use Gauss’s Law to figure out the charge inside them.
• The flux calculation considers both entering and leaving electric field lines.
• They provide a boundary to determine net electric flux and directly relate it to the enclosed charge.

By calculating the net flux, we can determine the difference between the electric field lines entering and leaving the surface. This net value is vital in finding the enclosed charge using Gauss's Law.

Enclosed Charge

The concept of enclosed charge is central to Gauss's Law. An enclosed charge refers to the total amount of electric charge contained within a closed surface. The relationship between net electric flux and enclosed charge is given by Gauss's Law:

• \( \Phi = \frac{Q_{enc}}{\varepsilon_0} \)

where \( Q_{enc} \) is the enclosed charge and \( \varepsilon_0 \) is the permittivity of free space, a constant valued at approximately \( 8.854 \times 10^{-12} \text{C}^2/\text{Nm}^2 \).
To find the enclosed charge from the net flux, you rearrange the formula:
\( Q_{enc} = \Phi \times \varepsilon_0 \).
This calculation reveals how much charge is inside the surface based on how electric field lines behave around that surface.
Understanding this helps explain electric behavior in closed systems, like how a charged particle's field affects surrounding materials. This explains why, in our original exercise, calculating the charge enclosed is possible by determining the net electric flux through a calculated difference of entering and leaving flux values.