Two identical circular loops, P and Q, each of radius r and carrying equal currents are kept in the parallel planes having a common axis passing through O. The direction of current in P is clockwise and in Q is anti-clockwise as seen from O which is equidistant from the loops P and Q. Find the magnitude of the net magnetic field at O

To find the magnitude of the net magnetic field at point O, we can use the principle of superposition.

Step 1: Determine the magnetic field produced by each individual loop at point O.

The magnetic field produced by a circular loop at its center is given by the formula:

B = (μ₀ * I * r²) / (2 * R³)

Where B is the magnetic field, μ₀ is the permeability of free space (4π × 10⁻⁷ T·m/A), I is the current in the loop, r is the radius of the loop, and R is the distance from the center of the loop to the point where the magnetic field is being measured (in this case, point O).

Since the loops P and Q are identical and equidistant from point O, the distance R is the same for both loops.

Step 2: Determine the direction of the magnetic field produced by each loop at point O.

The direction of the magnetic field produced by a current-carrying loop can be determined using the right-hand rule. For loop P, the current is clockwise, so the magnetic field at point O will be directed into the plane of the loop. For loop Q, the current is anti-clockwise, so the magnetic field at point O will be directed out of the plane of the loop.

Step 3: Apply the principle of superposition.

Since the magnetic field is a vector quantity, we can add the magnetic fields produced by each loop at point O using vector addition. Since the magnetic fields produced by loops P and Q are in opposite directions, we need to subtract the magnitude of the magnetic field produced by loop Q from the magnitude of the magnetic field produced by loop P.

Step 4: Calculate the net magnetic field at point O.

The net magnetic field at point O is given by:

B_net = B_P - B_Q

Where B_P is the magnetic field produced by loop P and B_Q is the magnetic field produced by loop Q.

Substituting the formula for the magnetic field into the equation, we have:

B_net = [(μ₀ * I * r²) / (2 * R³)] - [(μ₀ * I * r²) / (2 * R³)]

Simplifying the equation, we get:

B_net = 0

Therefore, the magnitude of the net magnetic field at point O is zero.