( according to page no. 14o of hc verma )figure shows a charge q placed at the centre of a hemisphere .A second charge Q is placed at the centre of a hemisphere .A second charge Q is placed at one of the positions A,B,C and D .in which position(s) of this second charge , the flux of the electric field through the hemisphere remains unchanged ?
a) A b) B c)C d) D



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To tackle the problem of electric flux through a hemisphere when a charge is placed at its center, we need to apply Gauss's Law, which states that the electric flux through a closed surface is proportional to the enclosed charge. Let's break down the scenario step by step.

Understanding Electric Flux

Electric flux (Φ) is defined as the product of the electric field (E) and the area (A) through which it passes, and it can be mathematically expressed as:

Φ = E · A · cos(θ)

Where θ is the angle between the electric field lines and the normal to the surface. In the case of a hemisphere, we are particularly interested in how the presence of additional charges affects the flux through the curved surface.

Analyzing the Setup

In this scenario, we have:

• A charge q located at the center of a hemisphere.
• A second charge Q that can be placed at one of four positions: A, B, C, or D.

Applying Gauss's Law

According to Gauss's Law, the electric flux through the hemisphere depends solely on the charge enclosed within the Gaussian surface. Since the charge q is at the center, it contributes to the electric field and thus the flux through the hemisphere.

Now, let's consider the effect of placing charge Q at different positions:

Position A

If charge Q is placed at position A (on the flat surface of the hemisphere), it does not affect the total enclosed charge within the hemisphere. Hence, the electric flux through the hemisphere remains unchanged.

Position B

Q at position B (also on the flat surface) similarly does not change the enclosed charge. Therefore, the flux remains the same as in position A.

Position C

When charge Q is placed at position C (inside the hemisphere but not at the center), it is still within the Gaussian surface. However, it contributes to the electric field lines that pass through the hemisphere, potentially altering the flux. Thus, the flux may change.

Position D

Finally, if charge Q is placed at position D (outside the hemisphere), it does not contribute to the electric field lines passing through the hemisphere at all. Therefore, the flux through the hemisphere remains unchanged, similar to positions A and B.

Conclusion on Flux Changes

In summary, the flux through the hemisphere remains unchanged when charge Q is placed at:

• Position A
• Position B
• Position D

However, it will change if charge Q is placed at position C. Thus, the correct answers are A, B, and D.