There is one circular loop of a uniform wire whose radius is R. Electric current I enters at one point on its circumference and leaves at a diametrically opposite point. What will be the net magnetic field intensity at the centre of the loop?

Options

A. `(mu_(0)i)/(4 r)`
B. `(mu_(0)i)/(2 r)`
C. `(mu_(0)i)/r `
D. Zero

Explanation

To find the net magnetic field intensity at the center of a circular loop of uniform wire with radius R, where electric current I enters at one point on its circumference and leaves at a diametrically opposite point, we can follow these steps:

### Step 1: Understand the Current Distribution
The current I enters the loop at point A and exits at point B, which is diametrically opposite to A. Since the current is uniformly distributed along the wire, the current flowing through each half of the loop will be I/2.

### Step 2: Analyze the Magnetic Field Contribution from Each Half
The circular loop can be divided into two halves:
- The upper half (from A to B)
- The lower half (from B to A)

Using the Biot-Savart law, we can determine the magnetic field at the center of the loop due to each half.

### Step 3: Calculate the Magnetic Field from the Upper Half
For the upper half of the loop, the magnetic field at the center (let's call it B1) can be calculated using the formula:
\[ B_1 = \frac{\mu_0 (I/2)}{4\pi R} \cdot \theta \]
where \(\theta\) is the angle subtended by the semicircular wire at the center, which is \(\pi\) radians for a semicircle. Thus,
\[ B_1 = \frac{\mu_0 (I/2)}{4\pi R} \cdot \pi = \frac{\mu_0 I}{8R} \]
The direction of this magnetic field (using the right-hand rule) will be into the plane of the loop.

### Step 4: Calculate the Magnetic Field from the Lower Half
For the lower half of the loop, the magnetic field at the center (let's call it B2) is similarly calculated:
\[ B_2 = \frac{\mu_0 (I/2)}{4\pi R} \cdot \theta = \frac{\mu_0 I}{8R} \]
However, the direction of this magnetic field will be out of the plane of the loop.

### Step 5: Determine the Net Magnetic Field
Now, we have:
- \(B_1\) (from the upper half) directed into the plane
- \(B_2\) (from the lower half) directed out of the plane

Since both magnetic fields have the same magnitude but opposite directions, they will cancel each other out:
\[ B_{\text{net}} = B_1 - B_2 = \frac{\mu_0 I}{8R} - \frac{\mu_0 I}{8R} = 0 \]

### Conclusion
The net magnetic field intensity at the center of the loop is zero.