Two identical conducting wires `AOB and COD` are placed at right angles to each other. The wire `AOB` carries an electric current `I_(1) and COD` carries a current `I_(2)`. The magnetic field on a point lying at a distance `d` from O, in a direction perpendicular to the plane of the wires `AOB and COD`, will be given by

Options

A. `(mu_(0))/(2pid)((I_(1))/(I_(2)))`
B. `(mu_(0))/(2pid)(I_(1)+I_(2))`
C. `(mu_(0))/(2pid)(I_(1)^(2)-I_(2)^(2))`
D. `(mu_(0))/(2pid)(I_(1)^(2)+I_(2)^(2))^(1//2)`

Answer

To find the magnetic field at a point P, which is at a distance d from the intersection point O of two perpendicular wires AOB and COD carrying currents I1 and I2 respectively, we can follow these steps:

### Step 1: Understand the Configuration
We have two identical conducting wires:
- Wire AOB carries current I1.
- Wire COD carries current I2.
These wires are placed at right angles to each other.

### Step 2: Calculate the Magnetic Field due to Wire AOB
The magnetic field (B1) at a distance d from a long straight wire carrying current I can be calculated using the formula:
\[ B = \frac{\mu_0 I}{2 \pi r} \]
where:
- \( \mu_0 \) is the permeability of free space,
- \( I \) is the current through the wire,
- \( r \) is the distance from the wire.

For wire AOB, the distance from point O to point P is d, so:
\[ B_1 = \frac{\mu_0 I_1}{2 \pi d} \]

### Step 3: Calculate the Magnetic Field due to Wire COD
Similarly, for wire COD, the magnetic field (B2) at the same distance d is given by:
\[ B_2 = \frac{\mu_0 I_2}{2 \pi d} \]

### Step 4: Determine the Direction of the Magnetic Fields
Using the right-hand rule:
- The magnetic field B1 due to wire AOB will be directed out of the plane (let's say in the positive z-direction).
- The magnetic field B2 due to wire COD will be directed into the plane (let's say in the negative z-direction).

### Step 5: Calculate the Resultant Magnetic Field
Since the two magnetic fields are perpendicular to each other, we can use the Pythagorean theorem to find the resultant magnetic field (B):
\[ B = \sqrt{B_1^2 + B_2^2} \]
Substituting the values of B1 and B2:
\[ B = \sqrt{\left(\frac{\mu_0 I_1}{2 \pi d}\right)^2 + \left(\frac{\mu_0 I_2}{2 \pi d}\right)^2} \]

### Step 6: Simplify the Expression
Factoring out common terms:
\[ B = \frac{\mu_0}{2 \pi d} \sqrt{I_1^2 + I_2^2} \]

### Final Answer
Thus, the magnetic field at point P is given by:
\[ B = \frac{\mu_0}{2 \pi d} \sqrt{I_1^2 + I_2^2} \]