A 90 cm long solenoid has six layers of windings of 450 turns each. If the diameter of solenoid is 2.2 cm and current carried is 6 A, then the magnitude of magnetic field inside the solenoid, near its centre is

Options

A. `50 pi G`
B. `60 pi G`
C. `72 pi G`
D. `80 pi G`

Explanation

To find the magnitude of the magnetic field inside a solenoid, we can use the formula:

\[ B = \mu_0 n I \]

where:
- \( B \) is the magnetic field,
- \( \mu_0 \) is the permeability of free space (\( 4\pi \times 10^{-7} \, \text{T m/A} \)),
- \( n \) is the number of turns per unit length,
- \( I \) is the current in amperes.

### Step 1: Calculate the total number of turns in the solenoid
Given that there are 6 layers of windings with 450 turns each:

\[ \text{Total turns} = \text{Number of layers} \times \text{Turns per layer} = 6 \times 450 = 2700 \, \text{turns} \]

**Hint:** Remember to multiply the number of layers by the number of turns in each layer to find the total number of turns.

### Step 2: Convert the length of the solenoid to meters
The length of the solenoid is given as 90 cm. To convert this to meters:

\[ \text{Length} = 90 \, \text{cm} = 0.90 \, \text{m} \]

**Hint:** Always convert measurements to standard units (meters for length) before using them in calculations.

### Step 3: Calculate the number of turns per unit length \( n \)
Using the total number of turns and the length of the solenoid:

\[ n = \frac{\text{Total turns}}{\text{Length}} = \frac{2700}{0.90} = 3000 \, \text{turns/m} \]

**Hint:** The number of turns per unit length is found by dividing the total number of turns by the length of the solenoid.

### Step 4: Substitute the values into the magnetic field formula
Now, we can substitute \( n \), \( I \), and \( \mu_0 \) into the formula for the magnetic field:

\[ B = \mu_0 n I = (4\pi \times 10^{-7}) \times (3000) \times (6) \]

**Hint:** Make sure to use the correct values for \( \mu_0 \), \( n \), and \( I \) when substituting into the formula.

### Step 5: Calculate \( B \)
Calculating the above expression:

\[ B = (4\pi \times 10^{-7}) \times (3000) \times (6) = 72\pi \times 10^{-4} \, \text{T} \]

### Step 6: Convert Tesla to Gauss
Since \( 1 \, \text{T} = 10^4 \, \text{Gauss} \):

\[ B = 72\pi \times 10^{-4} \, \text{T} = 72\pi \, \text{Gauss} \]

### Conclusion
Thus, the magnitude of the magnetic field inside the solenoid near its center is:

\[ B = 72\pi \, \text{Gauss} \]

**Final Answer:** The correct option is \( 72\pi \, \text{Gauss} \).