A thin "bar" magnet of length `'l'` and magnetic moment `'M'` is bent at the mid point so that the two parts are at right angles. The new magnetic length and magnetic moment are respectively

Options

A. `sqrt2l,sqrt2M`
B. `(l)/(sqrt2),(M)/(sqrt2)`
C. `sqrt2l,(M)/(sqrt2)`
D. `(l)/(sqrt2),sqrt2M`

Explanation

To solve the problem of a thin bar magnet that is bent at its midpoint, we need to determine the new magnetic length and magnetic moment after the bending. Here’s a step-by-step solution:

### Step 1: Understand the Initial Conditions
- The initial length of the bar magnet is denoted as \( l \).
- The magnetic moment \( M \) is given, which is defined as the product of the pole strength \( m \) and the effective length of the magnet.

### Step 2: Determine the Magnetic Moment
- The magnetic moment \( M \) can be expressed as:
\[ M = m \cdot l \]
where \( m \) is the pole strength.

### Step 3: Analyze the Bending of the Magnet
- The magnet is bent at its midpoint, creating two segments of length \( \frac{l}{2} \) each, forming a right angle (90 degrees) between them.

### Step 4: Calculate the New Magnetic Length
- The new magnetic length is the shortest distance between the north pole and the south pole after bending. This can be calculated using the Pythagorean theorem.
- The distance \( d \) can be calculated as:
\[ d = \sqrt{\left(\frac{l}{2}\right)^2 + \left(\frac{l}{2}\right)^2} \]
Simplifying this:
\[ d = \sqrt{\frac{l^2}{4} + \frac{l^2}{4}} = \sqrt{\frac{2l^2}{4}} = \sqrt{\frac{l^2}{2}} = \frac{l}{\sqrt{2}} \]
- Thus, the new magnetic length is:
\[ \text{New Magnetic Length} = \frac{l}{\sqrt{2}} \]

### Step 5: Calculate the New Magnetic Moment
- The pole strength \( m \) remains unchanged even after bending.
- The new magnetic moment \( M' \) can be expressed as:
\[ M' = m \cdot d \]
Substituting \( d \):
\[ M' = m \cdot \frac{l}{\sqrt{2}} \]
- Since \( m = \frac{M}{l} \), we can substitute this into the equation:
\[ M' = \frac{M}{l} \cdot \frac{l}{\sqrt{2}} = \frac{M}{\sqrt{2}} \]

### Final Results
- The new magnetic length is:
\[ \frac{l}{\sqrt{2}} \]
- The new magnetic moment is:
\[ \frac{M}{\sqrt{2}} \]

### Summary
- The new magnetic length is \( \frac{l}{\sqrt{2}} \).
- The new magnetic moment is \( \frac{M}{\sqrt{2}} \).