A current I flows in a thin wire in the shape of a regular polygon n sides. The magnetic induction at the centre of the polygon when `ntooo` is (R is the radius of its circumcircle)

Options

A. `(mu_(0)ni)/(2piR)"tan"(pi)/6`
B. `(mu_(0)n i)/(2piR) "tan"(pi)/R`
C. `(mu_(0)i)/(2R)`
D. zero

Answer

To find the magnetic induction (magnetic field) at the center of a regular polygon with \( n \) sides when \( n \) tends to infinity, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- Consider a regular polygon with \( n \) sides inscribed in a circle of radius \( R \) (the circumradius).
- The angle subtended by each side at the center \( O \) of the polygon is \( \frac{2\pi}{n} \).

2. **Identifying the Magnetic Field Contribution**:
- The magnetic field \( B \) at the center due to one side of the polygon can be calculated using the Biot-Savart law.
- For a finite length of wire, the magnetic field at a distance \( D \) from the wire carrying current \( I \) is given by:
\[ B = \frac{\mu_0 I}{4\pi D} \int \sin \theta \, dL \]
- Here, \( \theta \) is the angle between the wire and the line connecting the wire to the point where we measure the field.

3. **Calculating the Distance**:
- The distance \( D \) from the center to the midpoint of each side of the polygon is \( R \cos\left(\frac{\pi}{n}\right) \).

4. **Calculating the Total Magnetic Field**:
- The total magnetic field at the center due to all \( n \) sides can be expressed as:
\[ B = \frac{\mu_0 n I}{4\pi R \cos\left(\frac{\pi}{n}\right)} \left(\sin\left(\frac{\pi}{n}\right) + \sin\left(\frac{\pi}{n}\right)\right) \]
- This simplifies to:
\[ B = \frac{\mu_0 n I}{4\pi R \cos\left(\frac{\pi}{n}\right)} \cdot 2 \sin\left(\frac{\pi}{n}\right) \]

5. **Simplifying the Expression**:
- Further simplifying gives:
\[ B = \frac{\mu_0 n I}{2 R} \cdot \frac{\sin\left(\frac{\pi}{n}\right)}{\cos\left(\frac{\pi}{n}\right)} \]
- Using the identity \( \tan\left(\frac{\pi}{n}\right) \) gives:
\[ B = \frac{\mu_0 n I}{2 R} \tan\left(\frac{\pi}{n}\right) \]

6. **Taking the Limit as \( n \to \infty \)**:
- As \( n \) approaches infinity, \( \tan\left(\frac{\pi}{n}\right) \approx \frac{\pi}{n} \).
- Thus, we have:
\[ B \approx \frac{\mu_0 I}{2 R} \]

7. **Final Result**:
- Therefore, the magnetic induction at the center of the polygon when \( n \) tends to infinity is:
\[ B = \frac{\mu_0 I}{2 R} \]