Three rings, each having radius R, are placed mutually perpendicular to each other and each having its centre at the origin of co-ordinate system. If current I is flowing through each ring then the magnitude of the magnetic field at the common centre is:

Options

A. `sqrt(3)(mu_(0)I)/(2R)`
B. Zero
C. `(sqrt(2)-1)(mu_(0)I)/(2R)`
D. `(sqrt(3)-sqrt(2))(mu_(0)I)/(2R)`

Explanation

To find the magnitude of the magnetic field at the common center of three mutually perpendicular rings, each carrying a current \( I \), we can follow these steps:

### Step 1: Understand the Configuration
We have three rings, each with radius \( R \), placed mutually perpendicular to each other. The centers of all three rings are at the origin of the coordinate system.

### Step 2: Magnetic Field Due to a Single Ring
The magnetic field \( B \) at the center of a single ring carrying current \( I \) is given by the formula:
\[ B = \frac{\mu_0 I}{2R} \]
where \( \mu_0 \) is the permeability of free space.

### Step 3: Determine the Direction of Magnetic Fields
- For the first ring (let's say it lies in the XY-plane), the magnetic field at the center will be directed along the Z-axis.
- For the second ring (in the YZ-plane), the magnetic field will be directed along the X-axis.
- For the third ring (in the XZ-plane), the magnetic field will be directed along the Y-axis.

### Step 4: Write the Magnetic Fields
Let’s denote the magnetic fields due to the three rings as \( B_1 \), \( B_2 \), and \( B_3 \):
- \( B_1 = \frac{\mu_0 I}{2R} \hat{k} \) (along Z)
- \( B_2 = \frac{\mu_0 I}{2R} \hat{i} \) (along X)
- \( B_3 = \frac{\mu_0 I}{2R} \hat{j} \) (along Y)

### Step 5: Calculate the Magnitude of the Resultant Magnetic Field
The total magnetic field \( B \) at the center is the vector sum of the individual magnetic fields:
\[ B = B_1 + B_2 + B_3 \]
Since these fields are mutually perpendicular, we can use the Pythagorean theorem to find the magnitude:
\[ |B| = \sqrt{B_1^2 + B_2^2 + B_3^2} \]
Substituting the values:
\[ |B| = \sqrt{\left(\frac{\mu_0 I}{2R}\right)^2 + \left(\frac{\mu_0 I}{2R}\right)^2 + \left(\frac{\mu_0 I}{2R}\right)^2} \]
\[ |B| = \sqrt{3 \left(\frac{\mu_0 I}{2R}\right)^2} \]
\[ |B| = \frac{\mu_0 I}{2R} \sqrt{3} \]

### Step 6: Final Result
Thus, the magnitude of the magnetic field at the common center is:
\[ |B| = \frac{\mu_0 I \sqrt{3}}{2R} \]