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-z 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_(0)i)/(2R)`
B. `(mu_(0)i)/(4R)`
C. `(mu_(0)i)/(sqrt(2)R)`
D. `(mu_(0)i)/(2sqrt(2)R)`

Answer

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

### Step 1: Understand the Configuration
We have two semicircular loops:
- One semicircular loop lies in the x-y plane.
- The other semicircular loop lies in the x-z plane.
Both loops have the same radius \( R \) and carry the same current \( I \).

### Step 2: Determine the Magnetic Field from Each Semicircular Loop
The magnetic field \( B \) at the center of a semicircular loop carrying current \( I \) is given by the formula:
\[ B = \frac{\mu_0 I}{4R} \]
where \( \mu_0 \) is the permeability of free space.

### Step 3: Calculate the Magnetic Field from the Semicircle in the x-y Plane
For the semicircular loop in the x-y plane, the magnetic field \( B_{xy} \) at the center (common center) will point in the negative z-direction (using the right-hand rule):
\[ B_{xy} = \frac{\mu_0 I}{4R} \quad \text{(in the -z direction)} \]

### Step 4: Calculate the Magnetic Field from the Semicircle in the x-z Plane
For the semicircular loop in the x-z plane, the magnetic field \( B_{xz} \) at the center will point in the negative y-direction:
\[ B_{xz} = \frac{\mu_0 I}{4R} \quad \text{(in the -y direction)} \]

### Step 5: Combine the Magnetic Fields
The two magnetic fields \( B_{xy} \) and \( B_{xz} \) are perpendicular to each other. To find the resultant magnetic field \( B \), we can use the Pythagorean theorem:
\[ B = \sqrt{B_{xy}^2 + B_{xz}^2} \]
Substituting the values:
\[ B = \sqrt{\left(\frac{\mu_0 I}{4R}\right)^2 + \left(\frac{\mu_0 I}{4R}\right)^2} \]
\[ B = \sqrt{2 \left(\frac{\mu_0 I}{4R}\right)^2} \]
\[ B = \frac{\mu_0 I}{4R} \sqrt{2} \]

### Step 6: Final Result
Thus, the resultant magnetic field \( B \) at the common center of the two semicircular loops is:
\[ B = \frac{\mu_0 I}{2\sqrt{2} R} \]

### Conclusion
The resultant magnetic field at the common center of the two semicircular parts is:
\[ \frac{\mu_0 I}{2\sqrt{2} R} \]