A point charge `q_1` is placed inside cavity 1 and another point charge `q_2` is inside cavity 2. A point charge q is placed outside the conductor. For the situation described above, answer the following questions.
If charge `q_2` is at point Q (inside cavity 2), then `vecE` at the center of cavity 2 due to induced charge on the surface of cavity 2 would be

Options

A. `q_2//(4piepsilon_0r'_(2)^(2))` toward `q_2`
B. `q_2//(4piepsilon_0r_(2)^(2))` away from `q_2`
C. zero
D. cannot be determined

Explanation

To solve the problem, we need to analyze the situation step by step:

### Step 1: Understanding the Setup
We have a conductor with two cavities. Inside cavity 1, there is a point charge \( q_1 \), and inside cavity 2, there is another point charge \( q_2 \). Additionally, there is an external point charge \( q \) placed outside the conductor.

### Step 2: Induced Charges on the Cavity Surfaces
When a charge \( q_2 \) is placed inside cavity 2, it will induce a charge on the inner surface of cavity 2. The induced charge will be equal in magnitude but opposite in sign to \( q_2 \), thus it will be \( -q_2 \). However, this induced charge will not be uniformly distributed over the surface of cavity 2 due to the point nature of \( q_2 \).

### Step 3: Electric Field Due to Induced Charge
The electric field \( \vec{E} \) at any point inside a cavity due to induced charges on the surface of that cavity can be calculated using Gauss's law. However, because the induced charge on the surface of cavity 2 is non-uniform (due to the point charge \( q_2 \)), we cannot simply calculate the electric field at the center of cavity 2.

### Step 4: Conclusion
Since the induced charge on the surface of cavity 2 is non-uniform, the electric field \( \vec{E} \) at the center of cavity 2 cannot be determined. Therefore, the answer is that the electric field at the center of cavity 2 due to the induced charge on the surface of cavity 2 is not defined.

### Final Answer
The electric field \( \vec{E} \) at the center of cavity 2 due to the induced charge on the surface of cavity 2 is **undefined** (option D).

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