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

Answer

To calculate the magnetic susceptibility of the core of the solenoid, we can follow these steps:

### Step 1: Understand the relationship between magnetic induction (B), magnetic field strength (H), and magnetic permeability (μ).
The magnetic induction (B) in a solenoid can be expressed as:
\[ B = \mu H \]
where \( \mu \) is the permeability of the material and \( H \) is the magnetic field strength.

### Step 2: Calculate the magnetic field strength (H).
The magnetic field strength (H) in a solenoid is given by:
\[ H = N \cdot I \]
where \( N \) is the number of turns per unit length, and \( I \) is the current flowing through the solenoid.

Given:
- \( N = 10^3 \, \text{turns/m} \)
- \( I = 2 \, \text{A} \)

Substituting the values:
\[ H = 10^3 \cdot 2 = 2000 \, \text{A/m} \]

### Step 3: Use the given value of magnetic induction (B) to find the permeability (μ).
We know from the problem statement that:
\[ B = 4\pi \, \text{Wb/m}^2 \]

Using the relationship \( B = \mu H \):
\[ \mu = \frac{B}{H} \]

Substituting the values we have:
\[ \mu = \frac{4\pi}{2000} \]

### Step 4: Calculate the value of μ.
Using \( \pi \approx 3.14 \):
\[ \mu \approx \frac{4 \times 3.14}{2000} = \frac{12.56}{2000} = 0.00628 \, \text{Wb/(A·m)} \]

### Step 5: Relate permeability (μ) to the relative permeability (μr).
The permeability of free space (μ₀) is approximately:
\[ \mu_0 \approx 4\pi \times 10^{-7} \, \text{Wb/(A·m)} \]

The relative permeability (μr) is given by:
\[ \mu_r = \frac{\mu}{\mu_0} \]

Substituting the values:
\[ \mu_r = \frac{0.00628}{4\pi \times 10^{-7}} \]

### Step 6: Calculate μr.
Using \( \pi \approx 3.14 \):
\[ \mu_r \approx \frac{0.00628}{4 \times 3.14 \times 10^{-7}} \]
\[ \mu_r \approx \frac{0.00628}{1.256 \times 10^{-6}} \approx 5000 \]

### Step 7: Calculate the magnetic susceptibility (χ).
The magnetic susceptibility (χ) is related to the relative permeability (μr) as follows:
\[ \chi = \mu_r - 1 \]

Substituting the value of μr:
\[ \chi = 5000 - 1 = 4999 \]

### Final Answer:
The magnetic susceptibility of the core is:
\[ \chi = 4999 \]

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