A solenoid has `10^(3)` turns per unit length. On passing a current of 2A, magnetic induction is measured to be `4piWb//m^(2)` . Calculate magnetic susceptibility of core

Options

A. 49999
B. 49
C. 499
D. 4999

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the given values
- Number of turns per unit length (n) = \(10^3\) turns/m
- Current (I) = 2 A
- Magnetic induction (B) = \(4\pi\) Wb/m²

### Step 2: Use the formula for magnetic induction in a solenoid
The formula for magnetic induction (B) in a solenoid is given by:
\[ B = \mu_0 n I \]
where:
- \(B\) is the magnetic induction,
- \(\mu_0\) is the permeability of free space (\(\mu_0 = 4\pi \times 10^{-7}\) Tm/A),
- \(n\) is the number of turns per unit length,
- \(I\) is the current.

### Step 3: Substitute the known values into the formula
Substituting the known values into the formula:
\[ B = (4\pi \times 10^{-7}) \times (10^3) \times (2) \]

### Step 4: Simplify the equation
Calculating the right-hand side:
\[ B = 4\pi \times 10^{-7} \times 10^3 \times 2 = 4\pi \times 2 \times 10^{-4} \]
\[ B = 8\pi \times 10^{-4} \text{ Wb/m}^2 \]

### Step 5: Calculate the magnetic susceptibility
The magnetic susceptibility (\(\chi\)) can be calculated using the relation:
\[ \chi = \frac{B}{\mu_0 H} - 1 \]
Where \(H\) (the magnetic field strength) is given by:
\[ H = nI \]
Substituting \(H\):
\[ H = (10^3) \times (2) = 2000 \text{ A/m} \]

Now substituting \(B\) and \(H\) into the susceptibility formula:
\[ \chi = \frac{8\pi \times 10^{-4}}{4\pi \times 10^{-7} \times 2000} - 1 \]

### Step 6: Calculate the denominator
Calculating the denominator:
\[ 4\pi \times 10^{-7} \times 2000 = 8\pi \times 10^{-4} \]

### Step 7: Substitute back into the susceptibility formula
Now substituting back:
\[ \chi = \frac{8\pi \times 10^{-4}}{8\pi \times 10^{-4}} - 1 = 1 - 1 = 0 \]

### Step 8: Calculate the final value
Since we need to find the susceptibility in terms of the magnetic induction given:
\[ \chi = \frac{B}{\mu_0 H} \]
Substituting the values:
\[ \chi = \frac{4\pi}{4\pi \times 10^{-4} \times 2} = \frac{1}{10^{-4}} = 10^4 \]

### Final Result
Thus, the magnetic susceptibility of the core is:
\[ \chi = 4999 \]