Two identical short bar magnets, each having magnetic moment `M`, are placed a distance of `2d` apart with axes perpendicular to each other in a horizontal plane. The magnetic induction at a point midway between them is

Options

A. `(mu_(0))/(4pi)(sqrt2)(M)/(d^(3))`
B. `(mu_(0))/(4pi)(sqrt3)(M)/(d^(3))`
C. `((2mu_(0))/(4pi))(M)/(d^(3))`
D. `(mu_(0))/(4pi)(sqrt5)(M)/(d^(3))`

Explanation

To find the magnetic induction at a point midway between two identical short bar magnets, each having a magnetic moment \( M \) and placed a distance of \( 2d \) apart with their axes perpendicular to each other, we can follow these steps:

### Step 1: Understand the Configuration
We have two bar magnets positioned such that their centers are \( 2d \) apart. The axes of the magnets are perpendicular to each other. The point of interest (point P) is located midway between the two magnets, which means it is at a distance \( d \) from each magnet.

### Step 2: Determine the Magnetic Field Due to Each Magnet
For a bar magnet, the magnetic induction (magnetic field) at a point along the axial line (the line extending from the north to the south pole) is given by the formula:
\[ B_1 = \frac{2M \mu_0}{\pi d^3} \]
where \( B_1 \) is the magnetic induction due to the first magnet, \( M \) is the magnetic moment, \( \mu_0 \) is the permeability of free space, and \( d \) is the distance from the magnet.

For the second magnet, which is positioned perpendicular to the first, the magnetic field at point P (which is at a distance \( d \)) is given by the formula for the equatorial point:
\[ B_2 = \frac{M \mu_0}{4 \pi d^3} \]

### Step 3: Calculate the Magnetic Fields
1. **Magnetic Field from the First Magnet (Axial):**
\[ B_1 = \frac{2M \mu_0}{\pi d^3} \]

2. **Magnetic Field from the Second Magnet (Equatorial):**
\[ B_2 = \frac{M \mu_0}{4 \pi d^3} \]

### Step 4: Determine the Direction of the Magnetic Fields
- The magnetic field \( B_1 \) from the first magnet will be directed along the axis of the magnet.
- The magnetic field \( B_2 \) from the second magnet will be directed perpendicular to \( B_1 \).

### Step 5: Calculate the Resultant Magnetic Field
Since \( B_1 \) and \( B_2 \) are perpendicular to each other, we can find the net magnetic field \( B \) at point P using the Pythagorean theorem:
\[ B = \sqrt{B_1^2 + B_2^2} \]

Substituting the values of \( B_1 \) and \( B_2 \):
\[ B = \sqrt{\left(\frac{2M \mu_0}{\pi d^3}\right)^2 + \left(\frac{M \mu_0}{4 \pi d^3}\right)^2} \]

### Step 6: Simplify the Expression
Calculating \( B \):
\[ B = \sqrt{\frac{4M^2 \mu_0^2}{\pi^2 d^6} + \frac{M^2 \mu_0^2}{16 \pi^2 d^6}} \]
Factoring out common terms:
\[ B = \sqrt{\frac{M^2 \mu_0^2}{\pi^2 d^6} \left(4 + \frac{1}{16}\right)} \]
Calculating the term in the parentheses:
\[ 4 + \frac{1}{16} = \frac{64}{16} + \frac{1}{16} = \frac{65}{16} \]
Thus,
\[ B = \sqrt{\frac{M^2 \mu_0^2}{\pi^2 d^6} \cdot \frac{65}{16}} = \frac{M \mu_0}{\pi d^3} \cdot \sqrt{\frac{65}{16}} = \frac{M \mu_0}{4 \pi d^3} \sqrt{65} \]

### Final Answer
The magnetic induction at the point midway between the two magnets is:
\[ B = \frac{M \mu_0 \sqrt{65}}{4 \pi d^3} \]