Magnetic field at the centre of a circular loop with n turns is `0.50` mt. Calculate the value of n if current of `5.3` A flows in the loop and radius of loop is 2 cm.

Answer

To find the value of \( n \) (the number of turns) in a circular loop given the magnetic field at the center, we can use the formula for the magnetic field due to a circular coil. Here are the steps to solve the problem:

### Step 1: Write down the formula for the magnetic field at the center of a circular loop
The magnetic field \( B_0 \) at the center of a circular loop with \( n \) turns carrying current \( I \) is given by the formula:

\[ B_0 = \frac{\mu_0 n I}{2R} \]

where:
- \( B_0 \) is the magnetic field at the center,
- \( \mu_0 \) is the permeability of free space (\( 4\pi \times 10^{-7} \, \text{T m/A} \)),
- \( n \) is the number of turns,
- \( I \) is the current in amperes,
- \( R \) is the radius of the loop in meters.

### Step 2: Convert given values to appropriate units
Given:
- \( B_0 = 0.50 \, \text{mT} = 0.50 \times 10^{-3} \, \text{T} = 5 \times 10^{-4} \, \text{T} \)
- \( I = 5.3 \, \text{A} \)
- \( R = 2 \, \text{cm} = 0.02 \, \text{m} \)

### Step 3: Rearrange the formula to solve for \( n \)
We can rearrange the formula to solve for \( n \):

\[ n = \frac{B_0 \cdot 2R}{\mu_0 I} \]

### Step 4: Substitute the known values into the equation
Substituting the known values into the equation:

\[ n = \frac{(5 \times 10^{-4} \, \text{T}) \cdot (2 \cdot 0.02 \, \text{m})}{4\pi \times 10^{-7} \, \text{T m/A} \cdot 5.3 \, \text{A}} \]

### Step 5: Calculate the numerator and denominator
Calculating the numerator:

\[ \text{Numerator} = 5 \times 10^{-4} \cdot 2 \cdot 0.02 = 5 \times 10^{-4} \cdot 0.04 = 2 \times 10^{-5} \]

Calculating the denominator:

\[ \text{Denominator} = 4\pi \times 10^{-7} \cdot 5.3 \approx 4 \cdot 3.14 \cdot 10^{-7} \cdot 5.3 \approx 6.7 \times 10^{-6} \]

### Step 6: Calculate \( n \)
Now, substituting back into the equation for \( n \):

\[ n = \frac{2 \times 10^{-5}}{6.7 \times 10^{-6}} \approx 2.99 \approx 3 \]

### Final Answer
Thus, the value of \( n \) is approximately \( 3 \).

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