On connecting a battery to the two corners of a diagonal of a square conductor frame of side `a` the magnitude of the magnetic field at the centre will be

Options

A. Zero
B. `(mu_(0))/(pia)`
C. `(2mu_(0))/(pia)`
D. `(4mu_(0)i)/(pia)`

Answer

To find the magnitude of the magnetic field at the center of a square conductor frame when connected to a battery at two corners of a diagonal, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Configuration**:
- We have a square conductor frame with corners labeled A, B, C, and D. The battery is connected across the diagonal AC.

2. **Current Distribution**:
- When the battery is connected, current will flow through the sides AB, BC, CD, and DA. Since the sides are of equal length and made of the same material, the current will be equally divided among the two paths from A to C.

3. **Calculate Current in Each Segment**:
- Let the total current flowing from the battery be \( I \). Since the current splits equally, the current in each segment (AB, BC, CD, DA) will be \( I/2 \).

4. **Magnetic Field Due to Each Segment**:
- The magnetic field at the center of the square due to a straight conductor carrying current can be calculated using the formula:
\[ B = \frac{\mu_0 I}{4\pi r} \]
- Here, \( r \) is the distance from the wire to the point where the magnetic field is being calculated. For a square, the distance from the center to the midpoint of each side is \( \frac{a}{2\sqrt{2}} \).

5. **Calculate Magnetic Field Contribution from Each Side**:
- For each side of the square, the magnetic field at the center due to one side will be:
\[ B_{side} = \frac{\mu_0 (I/2)}{4\pi \left(\frac{a}{2\sqrt{2}}\right)} = \frac{\mu_0 I \sqrt{2}}{8\pi a} \]

6. **Direction of Magnetic Fields**:
- The magnetic field produced by the current in opposite sides of the square will be in opposite directions. For example, if the current flows in a clockwise direction, the magnetic field due to sides AB and CD will point in one direction, while the magnetic field due to sides BC and DA will point in the opposite direction.

7. **Net Magnetic Field Calculation**:
- Since the magnetic fields from opposite sides cancel each other out, the net magnetic field at the center of the square will be:
\[ B_{net} = B_{AB} + B_{CD} + B_{BC} + B_{DA} = 0 \]

### Final Answer:
The magnitude of the magnetic field at the center of the square conductor frame is **0**.