Magnitude of magnetic field (in SI units) at the centre of a hexagonal shape coil of side 10cm , 50 turns and carrying current I (Ampere) in units of `(mu_0I)/(pi)` is :

Options

A. `250 sqrt(3)`
B. `50sqrt(3)`
C. `500sqrt(3)`
D. `5sqrt(3)`

Explanation

To find the magnitude of the magnetic field at the center of a hexagonal coil with given parameters, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Parameters:**
- Side of the hexagon, \( a = 10 \, \text{cm} = 0.1 \, \text{m} \)
- Number of turns, \( N = 50 \)
- Current through the coil, \( I \)

2. **Determine the Distance from the Center to a Side:**
The distance from the center of the hexagon to the midpoint of one of its sides can be calculated using the formula:
\[ d = \frac{a}{2 \tan(30^\circ)} = \frac{0.1}{2 \cdot \frac{1}{\sqrt{3}}} = \frac{0.1 \sqrt{3}}{2} = \frac{0.1 \sqrt{3}}{2} \, \text{m} \]

3. **Calculate the Magnetic Field due to One Side:**
The magnetic field \( B \) at a distance \( d \) from a straight conductor carrying current \( I \) is given by the Biot-Savart Law:
\[ B = \frac{\mu_0 I}{4 \pi d} \]
Here, we will substitute \( d \) from the previous step:
\[ B_{\text{one side}} = \frac{\mu_0 I}{4 \pi \left(\frac{0.1 \sqrt{3}}{2}\right)} = \frac{\mu_0 I}{4 \pi} \cdot \frac{2}{0.1 \sqrt{3}} = \frac{\mu_0 I}{0.4 \pi \sqrt{3}} \]

4. **Calculate the Total Magnetic Field:**
Since there are 6 sides in a hexagonal coil, the total magnetic field at the center will be:
\[ B_{\text{total}} = 6 \cdot B_{\text{one side}} = 6 \cdot \frac{\mu_0 I}{0.4 \pi \sqrt{3}} = \frac{6 \mu_0 I}{0.4 \pi \sqrt{3}} = \frac{15 \mu_0 I}{\pi \sqrt{3}} \]

5. **Express in Terms of \(\frac{\mu_0 I}{\pi}\):**
To express the magnetic field in the required format:
\[ B_{\text{total}} = \frac{15}{\sqrt{3}} \cdot \frac{\mu_0 I}{\pi} \]
Simplifying \( \frac{15}{\sqrt{3}} = 5 \sqrt{3} \):
\[ B_{\text{total}} = 5 \sqrt{3} \cdot \frac{\mu_0 I}{\pi} \]

6. **Final Result:**
The magnitude of the magnetic field at the center of the hexagonal coil is:
\[ \frac{500 \sqrt{3} \mu_0 I}{\pi} \]