Question

Two concentric solenoids carry the same amount of current I in opposite rotational directions. The inner has a turns density n, and the outer has 4n. Find the magnetic field inside the inner solenoid in terms of the permeability mu.

Answer 1

Answer 2

The net magnetic field inside the inner solenoid has a magnitude of $3\mu_0 n I$, but it points in the opposite direction to the inner solenoid’s original field.

Answer 3

Magnetic Field Inside Concentric Solenoids:

Given:

• Inner solenoid: turns density = n, current = I (clockwise or counterclockwise)

• Outer solenoid: turns density = 4n, current = I (opposite direction)

• Permeability = μ

Key Idea:
The total magnetic field (B) inside the inner solenoid is the sum of the fields from both solenoids. Since the currents flow in opposite directions, their fields subtract.

Calculation:

1. Field from inner solenoid:
   $B_{\text{inner}}=\mu nI$

2. Field from outer solenoid:
   $B_{\text{outer}}=\mu (4n)I$ (but in the opposite direction).

3. 
   

Total Field Inside Inner Solenoid:
$B=B_{\text{inner}}-B_{\text{outer}}=\mu nI-\mu (4n)I=-3\mu nI$

The negative sign indicates the direction is determined by the outer solenoid's current.

Final Answer:

The magnetic field inside the inner solenoid is: