Two long coaxial solenoids each carry current I , but in opposite directions, as shown in Fig. 5.42. The inner solenoid (radius a) has turns per unit length, and the outer one (radius b) has .$n_2$Find B in each of the three regions: (i) inside the inner solenoid, (ii) between them, and (iii) outside both.

Write the expression for the magnetic field due to current carrying Solenoid.

$B={\mu }_{0}nl$for points inside the solenoid

$=0$for points outside the solenoid

Now, consider the three points A, B and C in the system of solenoid as shown in figure.

The point A is inside outer and inner solenoid.

The point B is outside of in between of outer and inner solenoid.

The point C is outside of outer solenoid.

Let’s consider that, radius of inner solenoid is $a$, $n_1$is the number of turns per unit length in the inner solenoid, $n_2$ number of turns per unit length in the inner solenoid and b is the radius of the outer solenoid.

Write the expression for the magnetic field at point A due to outer solenoid.

$B_1={\mu }_0{n}_{2}I$

Write the expression for the magnetic field at point A due to inner solenoid.

$B_1=-{\mu }_0{n}_{2}I$

Here, the negative sign indicates, the magnetic field due to inner solenoid in opposite direction to the direction of magnetic field due to outer solenoid.

Write the expression for total magnetic field at point A.

$B_A={\mu }_0\left(n_2-n_1\right)I$

Thus, the magnetic field at point which is inside the inner solenoid .$B_A={\mu }_0\left(n_2-n_1\right)I$

The magnetic field due to the inner solenoid is zero. As point B is outside of inner solenoid.

Write the expression for the magnetic field at point B due to outer solenoid.

$B={\mu }_0{n}_{2}I$

Write the expression for total magnetic field at point B.

$B_B={\mu }_0\left(n_2-n_1\right)I$

Thus, the magnetic field at point in between the outer and the inner solenoid .

$B_B={\mu }_0\left(n_2-n_1\right)I$

The magnetic field due to the inner solenoid is zero. As point C is outside the both solenoid.

Thus, the magnetic field at point which is outside ofthe inner and outer solenoid is zero.