Question

A closely wound solenoid 80 cm long has 5 layers of windings of 400 turns each. The diameter of the solenoid is 1.8 cm. If the current carried is 8.0 A, estimate the magnitude of B inside the solenoid near its centre:
${\text{A}}{\text{. 3}}{\text{.521}} \times {\text{1}}{{\text{0}}^{ - 3}}T \\\ {\text{B}}{\text{. 2}}{\text{.521}} \times {\text{1}}{{\text{0}}^{ - 3}}T \\\ {\text{C}}{\text{. 3}}{\text{.521}} \times {\text{1}}{{\text{0}}^{ - 5}}T \\\ {\text{D}}{\text{. 2}}{\text{.512}} \times {\text{1}}{{\text{0}}^{ - 2}}T \\\$

Answer 1

The magnetic field inside a solenoid is given as the product of permeability of core, number of turns in the coil per unit length and the current flowing through the solenoid. By using the given values, we can easily obtain the required value of the magnetic field inside the solenoid.

Complete answer:
We are given a closely wound solenoid. The length of the solenoid is given as
$l = 80cm = 0.8m$
We are given that the solenoid has 5 layers of windings and each of the windings have 400 turns. Therefore, the total number of turns in the coil can be calculated in the following way.
$n = 5 \times 400 = 2000$
The diameter of the solenoid is given as
$D = 1.8cm = 0.018m$
The amount of current which is flowing through the solenoid is given as
$I = 8A$
Now we know that the magnetic field inside a solenoid is given by the following expression.
$B = \dfrac{{{\mu _0}NI}}{l}$
Here ${\mu _0}$ signifies the permeability of the vacuum. Now we have all the required values given to us on the right hand side of the expression. Inserting all the known values, we can obtain the value of the magnetic field at the centre of the coil in the following way.
$B = \dfrac{{4\pi \times {{10}^{ - 7}} \times 2000 \times 8}}{{0.8}} = 2.512 \times {10^{ - 2}}T$
This is the required value of the magnetic field.

Hence, the correct answer is option D.

Note:
1. It should be noted that the term N/l in the formula for magnetic field signifies the number of turns in the coil per unit length of the solenoid.
2. The given solenoid has a core of air or vacuum so we have used permeability of vacuum. In case the core of solenoid is made of different material then we have permeability of that material instead of the permeability of the vacuum.