A` 0.5` m long conducting wire is bent in form of a circle, Calculate the magnetic field at the centre if wire carries a current of 5 A.

Answer

To calculate the magnetic field at the center of a circular wire of length 0.5 m carrying a current of 5 A, we can follow these steps:

### Step 1: Find the Radius of the Circular Wire
The length of the wire \( L \) is given as 0.5 m. The circumference \( C \) of a circle is given by the formula:
\[ C = 2\pi R \]
Where \( R \) is the radius of the circle. Since the wire is bent into a circle, we can equate the length of the wire to the circumference:
\[ L = 2\pi R \]
Rearranging this formula to find \( R \):
\[ R = \frac{L}{2\pi} = \frac{0.5}{2\pi} = \frac{0.5}{6.2832} \approx 0.0796 \, \text{m} \]

### Step 2: Use the Formula for Magnetic Field at the Center of a Circular Loop
The magnetic field \( B \) at the center of a circular loop carrying current \( I \) is given by the formula:
\[ B = \frac{\mu_0 I}{2R} \]
Where:
- \( \mu_0 \) is the permeability of free space, approximately \( 4\pi \times 10^{-7} \, \text{T m/A} \)
- \( I \) is the current in amperes
- \( R \) is the radius in meters

### Step 3: Substitute the Values into the Formula
Substituting the values we have:
- \( I = 5 \, \text{A} \)
- \( R \approx 0.0796 \, \text{m} \)
- \( \mu_0 = 4\pi \times 10^{-7} \, \text{T m/A} \)

Now, substituting these values into the formula:
\[ B = \frac{4\pi \times 10^{-7} \times 5}{2 \times 0.0796} \]

### Step 4: Calculate the Magnetic Field
Calculating the above expression:
\[ B = \frac{20\pi \times 10^{-7}}{0.1592} \approx \frac{62.8319 \times 10^{-7}}{0.1592} \approx 3.94 \times 10^{-5} \, \text{T} \]

### Final Answer
The magnetic field at the center of the circular wire is approximately:
\[ B \approx 3.94 \times 10^{-5} \, \text{T} \]

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