Two identical bar magnets, each of magnetic moment M, are placed as shown in figure. Magnetic field at point

Options

A. `2 ((mu _(0))/(4pi)) (M)/(d^(3))`
B. `((mu_(0))/(4pi))(M)/(d^(3))`
C. `sqrt5 ((mu_(0))/(4pi)) (M)/(d^(3))`
D. `2 sqrt5 ((mu_(0))/(4pi)) (M)/(d ^(3))`

Answer

To solve the problem of finding the magnetic field at point P due to two identical bar magnets, each with a magnetic moment \( M \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Configuration**:
- We have two identical bar magnets placed parallel to each other. Each magnet has a magnetic moment \( M \).
- The distance between the two magnets is \( 2D \), and the distance from each magnet to point P is \( D \).

2. **Magnetic Field due to a Single Bar Magnet**:
- The magnetic field \( B \) at a distance \( r \) from a magnetic dipole (bar magnet) with magnetic moment \( M \) is given by the formula:
\[ B = \frac{\mu_0}{4\pi} \cdot \frac{2M}{r^3} \]
- Here, \( \mu_0 \) is the permeability of free space.

3. **Calculate Magnetic Field from Each Magnet**:
- For the first magnet (let's call it Magnet A), the distance to point P is \( D \):
\[ B_A = \frac{\mu_0}{4\pi} \cdot \frac{2M}{D^3} \]
- For the second magnet (Magnet B), the distance to point P is also \( D \):
\[ B_B = \frac{\mu_0}{4\pi} \cdot \frac{2M}{D^3} \]

4. **Determine the Direction of the Magnetic Fields**:
- Since both magnets are aligned in the same direction, the magnetic fields \( B_A \) and \( B_B \) at point P will add up.

5. **Calculate the Total Magnetic Field at Point P**:
- The total magnetic field \( B_P \) at point P is the sum of the magnetic fields from both magnets:
\[ B_P = B_A + B_B = \frac{\mu_0}{4\pi} \cdot \frac{2M}{D^3} + \frac{\mu_0}{4\pi} \cdot \frac{2M}{D^3} \]
- This simplifies to:
\[ B_P = 2 \cdot \frac{\mu_0}{4\pi} \cdot \frac{2M}{D^3} = \frac{2\mu_0}{4\pi} \cdot \frac{2M}{D^3} \]

6. **Final Expression for the Magnetic Field**:
- Therefore, the magnetic field at point P is:
\[ B_P = \frac{2\mu_0 M}{2\pi D^3} = \frac{\mu_0 M}{\pi D^3} \]

7. **Select the Correct Option**:
- From the options provided, we can see that the correct answer matches with option A:
\[ B_P = 2 \cdot \frac{\mu_0 M}{4\pi D^3} \]