What is the value of total electric flux coming out of closed surface?

Explanation:

Gauss's Law and Electric Flux:

Definition: Gauss's law, a fundamental principle in electromagnetism, relates the electric flux passing through a closed surface to the net electric charge enclosed within that surface. It is mathematically expressed as:

Where:

• : Represents the total electric flux through the closed surface.
• : Is the net charge enclosed within the surface.
• : Is the permittivity of free space, a constant.

Key Idea: The total electric flux through a closed surface depends solely on the charge enclosed within that surface. If no net charge is enclosed, the total flux is zero.

Correct Option Analysis:

The correct answer to the question is:

Option 1: Zero

This is because when the net charge enclosed by a closed surface is zero, the total electric flux through that surface is also zero, as per Gauss's Law. This happens regardless of the distribution of charges outside the surface or the shape of the surface itself. The electric field lines entering the surface will equal those exiting, resulting in no net flux.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 2: Equal to volume charge density

This option is incorrect because Gauss's law relates the total electric flux to the net charge enclosed by the surface, not directly to the volume charge density. The volume charge density () is used to calculate the total charge enclosed () by integrating it over the volume:

However, the flux is related to the total charge and not directly to the density.

Option 3: Equal to the total charge enclosed by the surface

This statement is partially correct but misrepresents the relationship. The total electric flux is not equal to the charge enclosed; instead, it is proportional to the enclosed charge and is given by:

The proportionality constant is , the permittivity of free space.

Option 4: Equal to the surface charge density

This option is also incorrect because Gauss's law concerns the net charge enclosed by the surface, not the surface charge density. Surface charge density () is a local property and does not directly dictate the total electric flux unless integrated over the surface to find the total charge.

Conclusion:

Gauss's law provides a powerful framework for understanding electric flux and its relation to enclosed charge. The total electric flux coming out of a closed surface is zero if the net charge enclosed within the surface is zero. This principle is fundamental in electromagnetism and has applications in various fields, including electrostatics and electric field analysis.