Magnetic Field at Midway between Two Circular Rings Carrying Current

When two circular rings carrying current are placed at a distance R apart, we can determine the magnetic field at the midway point between the rings. To do so, we need to consider the contributions from both rings individually.

Contributions from the First Ring:
The magnetic field at a point due to a current-carrying circular loop can be calculated using Ampere's law. At the midway between the rings, the magnetic field due to the first ring will be directed perpendicular to the line joining the center of the ring to the midway point. This magnetic field is given by:

B1 = (μ0 * I * R^2) / (2 * (R^2 + (R/2)^2)^(3/2))

where μ0 is the permeability of free space, I is the current, and R is the radius of the ring.

Contributions from the Second Ring:
Similarly, the magnetic field at the midway point due to the second ring will also be directed perpendicular to the line joining the center of the ring to the midway point. This magnetic field is given by the same expression as for the first ring:

B2 = (μ0 * I * R^2) / (2 * (R^2 + (R/2)^2)^(3/2))

Total Magnetic Field at Midway:
To find the total magnetic field at the midway point, we need to consider the contributions from both rings. Since the rings are placed R distance apart, the total magnetic field at the midway point will be the vector sum of the magnetic fields due to each ring.

Btotal = B1 + B2

Explanation:
The magnetic field at the midway point between two circular rings carrying current can be calculated by considering the contributions from each ring separately. The magnetic field due to each ring can be determined using Ampere's law, which relates the magnetic field to the current enclosed by a closed loop.

By applying Ampere's law and considering the geometry of the problem, we can calculate the magnetic field due to each ring. The magnetic field at the midway point will be directed perpendicular to the line joining the center of the ring to the midway point.

To find the total magnetic field at the midway point, we need to sum up the contributions from both rings. Since the rings are placed R distance apart, the magnetic fields due to each ring will add up at the midway point. The total magnetic field at the midway point can be calculated by taking the vector sum of the magnetic fields due to each ring.

It is important to note that the direction of the magnetic field can be determined using the right-hand rule. By curling the fingers of your right hand in the direction of the current, the thumb will point in the direction of the magnetic field.