The current on the winding of a toroid is 2 A. It has 400 turns and mean circumferential length is 40 cm. With the help of search coil and charge measuring instrument the magnetic field is found to be 1 T. The susceptibility is

Options

A. 2000
B. 2500
C. 1000
D. 1500

Answer

To find the susceptibility of the toroid, we will follow these steps:

### Step 1: Identify the given values
- Current (I) = 2 A
- Number of turns (N) = 400
- Mean circumferential length (l) = 40 cm = 0.4 m
- Magnetic field (B) = 1 T

### Step 2: Calculate the number of turns per unit length (n)
The number of turns per unit length (n) can be calculated using the formula:
\[ n = \frac{N}{l} \]
Substituting the values:
\[ n = \frac{400 \text{ turns}}{0.4 \text{ m}} = 1000 \text{ turns/m} \]

### Step 3: Use the formula for the magnetic field in a medium
The magnetic field (B) in a medium is given by:
\[ B = \mu_0 \mu_r n I \]
Where:
- \( \mu_0 \) (permeability of free space) = \( 4\pi \times 10^{-7} \, \text{T m/A} \)
- \( \mu_r \) (relative permeability)
- \( n \) = number of turns per unit length
- \( I \) = current

### Step 4: Rearrange the formula to solve for \( \mu_r \)
Rearranging the equation gives:
\[ \mu_r = \frac{B}{\mu_0 n I} \]

### Step 5: Substitute the known values into the equation
Substituting the values:
\[ \mu_r = \frac{1 \, \text{T}}{(4\pi \times 10^{-7} \, \text{T m/A}) \times (1000 \, \text{turns/m}) \times (2 \, \text{A})} \]

### Step 6: Calculate \( \mu_r \)
Calculating the denominator:
\[ \mu_0 n I = (4\pi \times 10^{-7}) \times 1000 \times 2 \]
\[ = 8\pi \times 10^{-4} \, \text{T} \]

Now substituting this back into the equation for \( \mu_r \):
\[ \mu_r = \frac{1}{8\pi \times 10^{-4}} \]

### Step 7: Calculate \( \mu_r \)
Using \( \pi \approx 3.14 \):
\[ \mu_r \approx \frac{1}{8 \times 3.14 \times 10^{-4}} \]
\[ \mu_r \approx \frac{1}{2.5136 \times 10^{-3}} \]
\[ \mu_r \approx 397.88 \]

### Step 8: Calculate the susceptibility \( \zeta \)
The susceptibility \( \zeta \) is given by:
\[ \zeta = \mu_r - 1 \]
Substituting the value of \( \mu_r \):
\[ \zeta = 397.88 - 1 \]
\[ \zeta \approx 396.88 \]

### Final Answer
The susceptibility of the toroid is approximately \( 396.88 \).

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