Question

A closely wound solenoid $80 \mathrm{~cm}$ long has 5 layers of windings of 400 turns each. The diameter of the solenoid is $1.8 \mathrm{~cm} .$ If the current carried is $8.0 \mathrm{~A}$, estimate the magnitude of $\mathrm{B}$ inside the solenoid near its centre.

Answer 1

Hint: Inside a solenoid, the magnetic field is proportional to both the current being applied and the number of turns per unit length. There is no dependence on the solenoid's diameter, and the strength of the field does not depend on the position inside the solenoid, i.e. the field is constant inside.

Formula used:
$\mathrm{B}=\mu_{\mathrm{o}} \mathrm{nI}$

Complete Step-by-Step solution:
A solenoid is a device consisting of a long conducting wire that is composed of many closely packed wire loops. The magnetic field lines are approximately uniform and parallel to each other inside the solenoid.
Length of solenoid $\mathrm{L}=80 \mathrm{~cm}=0.8 \mathrm{~m}$
Diameter of solenoid $\mathrm{d}=1.8 \mathrm{~cm}$

As $\mathrm{d}<<\mathrm{L}$, thus we can treat the solenoid as solenoid of infinite length w.r.t its diameter.

Current flowing through the solenoid $\mathrm{I}=8.0 \mathrm{~A}$

Total number of turns $\mathrm{N}=5 \times 400=2000$

Number of turns per unit length $\mathrm{n}=\dfrac{\mathrm{N}}{\mathrm{L}}$ $\therefore \mathrm{n}=\dfrac{2000}{0.8}=2500$

Magnetic field at the center of solenoid $\mathrm{B}=\mu_{\mathrm{o}} \mathrm{nI}$
$\therefore \mathrm{B}=4 \pi \times 10^{-7} \times 2500 \times 8.0$

We get $\mathrm{B}=2.5 \times 10^{-2} \mathrm{~T}$

Additional Information :
The change in magnetic field energy can be used as the rod is inserted into the solenoid to estimate approximately the force with which a solenoid pulls on ferromagnetic rods placed near it The energy density of the magnetic field depends on the strength of the field, the square strength, and also on the magnetic permeability of the material it fills. Iron has a permeability much much greater than a vacuum. The forces of a few Newtons can be exerted even by small solenoids.

Note:
A solenoid is a long wire coil wrapped in numerous turns. It creates a nearly uniform magnetic field inside when a current passes through it. Solenoids can convert mechanical action to electrical current, and are thus very commonly used as switches. The field within a solenoid is. A Directly proportional to its duration. B. Straight proportional to current.