Two identical thin bar magnets, each of length L and pole strength m are placed at right angles to each other, with the N pole of one touching the S-pole of the other. Find the magnetic moment of the system.

Explanation

To find the magnetic moment of the system formed by two identical thin bar magnets placed at right angles to each other, we can follow these steps:

### Step 1: Understand the Configuration
We have two identical bar magnets, each with length \( L \) and pole strength \( m \). One magnet is oriented horizontally, and the other is oriented vertically, with the north pole of the horizontal magnet touching the south pole of the vertical magnet.

### Step 2: Identify the Effective Length
The magnetic moment \( \mu \) of a magnet is given by the product of its pole strength and the effective distance between the poles. For our configuration, the effective distance between the two poles (N pole of one magnet and S pole of the other) can be determined using the Pythagorean theorem.

### Step 3: Calculate the Effective Distance
Since the two magnets are at right angles to each other, the effective distance \( d \) between the north pole of the vertical magnet and the south pole of the horizontal magnet can be calculated as follows:

\[ d = \sqrt{L^2 + L^2} = \sqrt{2L^2} = L\sqrt{2} \]

### Step 4: Calculate the Magnetic Moment
The magnetic moment \( \mu \) of the system can be calculated using the formula:

\[ \mu = m \times d \]

Substituting the effective distance we found:

\[ \mu = m \times (L\sqrt{2}) = mL\sqrt{2} \]

### Step 5: Consider the Contribution of Both Magnets
Since we have two magnets, we need to consider the contribution of both. The total magnetic moment of the system is:

\[ \mu_{\text{total}} = mL\sqrt{2} + mL\sqrt{2} = 2mL\sqrt{2} \]

### Final Answer
Thus, the magnetic moment of the system is:

\[ \mu_{\text{total}} = 2mL\sqrt{2} \]

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