A "bar" magnet of moment `M` is cut into two identical pieces along the length. One piece is bent in the form of a semi circle. If two pieces are perpendicular to each other, then resultant magnetic moment is

Options

A. `((M)/(R ))^(2)+((M)/(2))^(2)`
B. `sqrt(((M)/(R ))^(2)+((M)/(2))^(2))`
C. `sqrt(((M)/(R ))^(2)-((M)/(2))^(2))`
D. `(M)/(pi)+(M)/(2)`

Explanation

To solve the problem step by step, we will analyze the situation involving the bar magnet and its resultant magnetic moment after being cut and bent.

### Step 1: Understand the Initial Magnetic Moment
The magnetic moment of the original bar magnet is given as \( M \). The length of the magnet is denoted as \( L \).

### Step 2: Cut the Magnet
When the bar magnet is cut into two identical pieces along its length, each piece will have:
- Magnetic moment \( M_1 = \frac{M}{2} \)
- Length \( L \) remains the same for each piece.

### Step 3: Bend One Piece into a Semicircle
One of the pieces is bent into the shape of a semicircle. The pole strength of this semicircular piece remains \( \frac{M}{2} \).

### Step 4: Determine the Radius of the Semicircle
The length of the semicircle is equal to the original length of the magnet, \( L \). The length of a semicircle is given by:
\[ L = \pi R \]
From this, we can find the radius \( R \):
\[ R = \frac{L}{\pi} \]

### Step 5: Calculate the Magnetic Moment of the Semicircular Piece
The distance between the north and south poles of the semicircular magnet is the diameter, which is:
\[ 2R = 2 \times \frac{L}{\pi} = \frac{2L}{\pi} \]
The magnetic moment \( M_1 \) of the semicircular piece can be calculated as:
\[ M_1 = \text{Pole Strength} \times \text{Distance} = \left(\frac{M}{2}\right) \times \left(\frac{2L}{\pi}\right) = \frac{ML}{\pi} \]

### Step 6: Calculate the Magnetic Moment of the Straight Piece
The other piece remains straight and has a magnetic moment \( M_2 \):
\[ M_2 = \text{Pole Strength} \times \text{Length} = \left(\frac{M}{2}\right) \times L = \frac{ML}{2} \]

### Step 7: Resultant Magnetic Moment Calculation
Since the two pieces are perpendicular to each other, we can use the formula for the resultant of two perpendicular vectors:
\[ M_{\text{resultant}} = \sqrt{M_1^2 + M_2^2} \]
Substituting the values of \( M_1 \) and \( M_2 \):
\[ M_{\text{resultant}} = \sqrt{\left(\frac{ML}{\pi}\right)^2 + \left(\frac{ML}{2}\right)^2} \]

### Step 8: Simplifying the Resultant
Calculating the squares:
\[ M_{\text{resultant}} = \sqrt{\frac{M^2L^2}{\pi^2} + \frac{M^2L^2}{4}} = \sqrt{M^2L^2\left(\frac{1}{\pi^2} + \frac{1}{4}\right)} \]
Factoring out \( M^2L^2 \):
\[ M_{\text{resultant}} = ML \sqrt{\frac{1}{\pi^2} + \frac{1}{4}} \]

### Step 9: Final Expression
Thus, the resultant magnetic moment is:
\[ M_{\text{resultant}} = ML \sqrt{\frac{1}{\pi^2} + \frac{1}{4}} \]