Two short bar magnets of magnetic moments 'M' each are arranged at the opposite corners of a square of side 'd' such that their centre coincide with the corners and their axes are parallel to one side of the square. If the like poles are in the same direction, the magnetic induction at any of the other corners of the square is

Options

A. `(mu_(0))/(4pi) (M)/(d^(3))`
B. `(mu_(0))/(4pi) (2M)/(d ^(3))`
C. `(mu_(0))/(2pi) (M)/(d ^(3))`
D. `(mu_(0))/(2pi) (2M)/(d ^(3))`

Answer

To solve the problem of finding the magnetic induction at one of the corners of the square due to two short bar magnets with magnetic moments \( M \), we can follow these steps:

### Step 1: Understanding the setup
We have two bar magnets positioned at opposite corners of a square with side length \( d \). The centers of the magnets coincide with the corners, and their axes are parallel to one side of the square. The like poles of the magnets are facing each other.

### Step 2: Identify the corners
Let's label the corners of the square as \( A, B, C, \) and \( D \). Assume the magnets are positioned at corners \( A \) and \( C \). We want to find the magnetic induction at corner \( B \) (or equivalently at corner \( D \)).

### Step 3: Determine the magnetic field contributions
1. **Magnetic field due to magnet at corner \( A \)**:
- The magnetic field at point \( B \) due to the magnet at \( A \) is along the axis of the magnet. This is an axial point for magnet \( A \).
- The formula for the magnetic field \( B_1 \) at an axial point is given by:
\[ B_1 = \frac{\mu_0}{4\pi} \cdot \frac{2M}{d^3} \]

2. **Magnetic field due to magnet at corner \( C \)**:
- The magnetic field at point \( B \) due to the magnet at \( C \) is in the opposite direction since the like poles are facing each other. This is an equatorial point for magnet \( C \).
- The formula for the magnetic field \( B_2 \) at an equatorial point is given by:
\[ B_2 = \frac{\mu_0}{4\pi} \cdot \frac{M}{d^3} \]

### Step 4: Calculate the net magnetic field
Since the magnetic fields \( B_1 \) and \( B_2 \) are in opposite directions, we can find the net magnetic field \( B \) at point \( B \) by subtracting \( B_2 \) from \( B_1 \):
\[ B = B_1 - B_2 \]
Substituting the expressions for \( B_1 \) and \( B_2 \):
\[ B = \frac{\mu_0}{4\pi} \cdot \frac{2M}{d^3} - \frac{\mu_0}{4\pi} \cdot \frac{M}{d^3} \]
\[ B = \frac{\mu_0}{4\pi} \cdot \left(\frac{2M}{d^3} - \frac{M}{d^3}\right) \]
\[ B = \frac{\mu_0}{4\pi} \cdot \frac{M}{d^3} \]

### Final Answer
Thus, the magnetic induction at any of the other corners of the square is:
\[ B = \frac{\mu_0}{4\pi} \cdot \frac{M}{d^3} \]