A current loop consists of two identical semicircular parts each of radius R, one lying in the x-y plane and the other in x-y plane. If the current in the loop is i, the resultant magnetic field due to two semicircular parts at their common centre is

Options

A. `(mu_0i)/(2sqrt2R)`
B. `(mu_0i)/(2R)`
C. `(mu_0i)/(4R)`
D. `(mu_0i)/(sqrt2R)`

Explanation

To solve the problem of finding the resultant magnetic field at the common center of two identical semicircular current loops, we will follow these steps:

### Step 1: Understanding the Configuration
We have two semicircular loops, each of radius \( R \). One loop lies in the x-y plane, and the other lies in the x-z plane. The current \( i \) flows through both loops.

### Step 2: Determine the Magnetic Field due to Each Semicircular Loop
Using the formula for the magnetic field at the center of a semicircular loop of radius \( R \) carrying a current \( i \):
\[ B = \frac{\mu_0 i}{4R} \]
where \( \mu_0 \) is the permeability of free space.

### Step 3: Calculate the Magnetic Field Direction
1. **For the semicircular loop in the x-y plane (B1)**:
- By the right-hand rule, if the current flows counterclockwise when viewed from above, the magnetic field \( B_1 \) will point in the positive z-direction.
- Therefore, \( B_1 = \frac{\mu_0 i}{4R} \hat{k} \).

2. **For the semicircular loop in the x-z plane (B2)**:
- Again, using the right-hand rule, if the current flows counterclockwise when viewed from the positive y-axis, the magnetic field \( B_2 \) will point in the negative y-direction.
- Therefore, \( B_2 = -\frac{\mu_0 i}{4R} \hat{j} \).

### Step 4: Resultant Magnetic Field Calculation
The resultant magnetic field \( B_{net} \) at the common center is the vector sum of \( B_1 \) and \( B_2 \):
\[ B_{net} = B_1 + B_2 = \frac{\mu_0 i}{4R} \hat{k} - \frac{\mu_0 i}{4R} \hat{j} \]

### Step 5: Final Result
Thus, the resultant magnetic field at the common center due to the two semicircular parts is:
\[ B_{net} = \frac{\mu_0 i}{4R} \hat{k} - \frac{\mu_0 i}{4R} \hat{j} \]