Find the magnitude of the magnetic fieldat the center of an equilateral triangular loop of side length 1m which is carrying a current of 10A. ( Take `mu_(0) = 4pi xx 10^(-7) NA^(-2))`

Options

A. `3muT`
B. `9muT`
C. `1muT`
D. `18 muT`

Explanation

To find the magnitude of the magnetic field at the center of an equilateral triangular loop carrying a current, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Side length of the triangle (a) = 1 m
- Current (I) = 10 A
- Permeability of free space (μ₀) = \(4\pi \times 10^{-7} \, \text{NA}^{-2}\)

2. **Determine the Distance from the Center to a Side (r)**:
- For an equilateral triangle, the distance from the center to a side (r) can be calculated using the formula:
\[ r = \frac{a}{2\sqrt{3}} \]
- Substituting the value of a:
\[ r = \frac{1}{2\sqrt{3}} \approx 0.2887 \, \text{m} \]

3. **Use the Formula for Magnetic Field at the Center of an Equilateral Triangle**:
- The formula for the magnetic field (B) at the center of an equilateral triangular loop is given by:
\[ B = \frac{3\mu_0 I}{4\pi r} \left( \sin 60^\circ + \sin 60^\circ \right) \]
- Since \(\sin 60^\circ = \frac{\sqrt{3}}{2}\), we can simplify:
\[ B = \frac{3\mu_0 I}{4\pi r} \left( 2 \cdot \frac{\sqrt{3}}{2} \right) = \frac{3\mu_0 I \sqrt{3}}{4\pi r} \]

4. **Substitute the Values into the Formula**:
- Now substituting the values of μ₀, I, and r into the equation:
\[ B = \frac{3 \times (4\pi \times 10^{-7}) \times 10 \times \sqrt{3}}{4\pi \times \left(\frac{1}{2\sqrt{3}}\right)} \]

5. **Simplify the Expression**:
- The \(4\pi\) in the numerator and denominator cancels out:
\[ B = \frac{3 \times 10^{-7} \times 10 \times \sqrt{3}}{\frac{1}{2\sqrt{3}}} \]
- This simplifies to:
\[ B = 3 \times 10^{-7} \times 10 \times 2 \times 3 = 18 \times 10^{-6} \, \text{T} \]

6. **Convert to Microtesla**:
- Since \(1 \, \text{T} = 10^6 \, \mu\text{T}\):
\[ B = 18 \, \mu\text{T} \]

### Final Answer:
The magnitude of the magnetic field at the center of the equilateral triangular loop is **18 microtesla**.