Question

one solenoid is inside the other solenoid. both solenoids have length l and n turns. the radius of the inner solenoid is 2r and the radius of the outer solenoid is 3r. the inner solenoid carries current 4i and the outer solenoid carries current i in opposite directions as shown. what is the magnetic field at a distance r from the central axis? both solenoids are long and thin, i.e., l >> r.

Answer 1

The net magnetic field at a distance r from the central axis, with one solenoid inside the other with opposite currents, is three times the magnetic field created by the outer solenoid alone, assuming an infinite solenoid approximation.

Step-by-step explanation:

To find the magnetic field at a distance r from the central axis when one solenoid is inside another, we need to use Ampere's law and the superposition principle for magnetic fields. Since the current directions are opposite in the two solenoids, their magnetic fields will be in opposite directions as well.

For the inner solenoid with radius 2r and current 4i, the magnetic field inside it (using the approximation for an infinite solenoid) is given by B1 = μ0*(n/l)*4i, where μ0 is the permeability of free space. Similarly, for the outer solenoid with radius 3r and current i, the magnetic field inside it is B2 = μ0*(n/l)*i.

Because the question asks for the magnetic field at a distance r from the central axis, which lies inside both solenoids, we find that the net magnetic field is Bnet = B1 - B2 = μ0*(n/l)*(4i - i), which simplifies to Bnet = μ0*3(n/l)*i.

Answer 2

To find the magnetic field at a distance r from the central axis of the solenoids, use the formula B = mu0 * n * I. For the inner solenoid, the field is 4(mu0 * 4n * 4i) / (2 * 2r). For the outer solenoid, the field is mu0 * n * i / (3r).

Step-by-step explanation:

To find the magnetic field at a distance r from the central axis of the solenoid, we can use the formula for the magnetic field inside a solenoid:

B = μ0 * n * I

Where μ0 is the permeability of free space, n is the number of turns per unit length, and I is the current. In the case of the inner solenoid, n = 4n (since it has 4 times the current of the outer solenoid) and the radius is 2r. So, the magnetic field at a distance r from the central axis of the inner solenoid is 4(μ0 * 4n * 4i) / (2 * 2r).

For the outer solenoid, n = n and the radius is 3r. So, the magnetic field at a distance r from the central axis of the outer solenoid is μ0 * n * i / (3r).