Question

The magnetic field $B$ inside a long solenoid, carrying a current of $5.00{\text{ A}}$ , is $3.14 \times {10^{ - 2}}{\text{ T}}$ . Find the number of turns per unit length of the solenoid.

Answer 1

Hint : The magnetic field of a long solenoid is even inside and is independent of the location. The magnetic field is directly proportional to the current flowing through the coils of the solenoid, and the number of turns per unit length of the solenoid.

Formula used: In this solution we will be using the following formula;
$B = \mu nI$ where $B$ is the magnetic field inside a solenoid, $\mu$ is the permeability of free space, $n$ is the number of turns per unit length, and finally $I$ is the current through the coils of the solenoid

Complete step by step answer
Whenever current is flowing through a long straight conductor, a magnetic field is generated around the conductor given by the right hand rule. This magnetic field varies according to the distance from the conductor. It reduces as we move further away from the conductor. But when this conductor is coiled around into what we call a solenoid, the magnetic field cancels out everywhere but the volume encompassed by the solenoid. The magnetic field in the solenoid is considered uniform i.e. non varying and the magnetic lines are equally spaced. The magnetic field inside a solenoid is given as
$B = \mu nI$ where $B$ is the magnetic field inside a solenoid, $\mu$ is the permeability of free space, $n$ is the number of turns per unit length, and finally $I$ is the current through the coils of the solenoid
Hence, to calculate $n$ we rearrange as in
$n = \dfrac{B}{{\mu I}}$
Hence, by inserting all known values, we have
$n = \dfrac{{3.14 \times {{10}^{ - 2}}}}{{4\pi \times {{10}^{ - 7}} \times 5.00}}$ . By computation
$n = 4997{\text{ turns/metre}}$
$\therefore n = 4997{m^{ - 1}}$ .

Note
In actuality, the magnetic field inside a solenoid isn’t completely uniform and varies depending on location. It is often weaker at the ends, but strongest at the centre. Also, the magnetic field doesn’t completely cancel out outside the solenoid but reduces dramatically (exponentially) as we move away from the solenoid. These two effects are usually neglected under most situations since there is so little.