Question

A solenoid of length 0.5 m has a radius of 1 cm and is made up of 'm' number of turns. It carries a current of 5 A. If the magnitude of the magnetic field inside the solenoid is $6.28 \times 10^{-3}$ T, then the value of m is:

Answer 1

Correct Answer: 500

The magnetic field inside a solenoid is given by:

$$B = \mu_0 n i,$$

where:
- $B = 6.28 \times 10^{-3} \, \text{T}$ is the magnetic field,
- $\mu_0 = 4\pi \times 10^{-7} \, \text{T m/A}$ is the permeability of free space,
- $n = \frac{m}{\ell}$ is the number of turns per unit length,
- $i = 5 \, \text{A}$ is the current,
- $\ell = 0.5 \, \text{m}$ is the length of the solenoid.

Step 1: Rearranging the Formula
Substituting the given values:

$$\mu_0 \left( \frac{m}{\ell} \right) i = B.$$

Rearranging to find $m$:

$$m = \frac{B \ell}{\mu_0 i}.$$

Step 2: Substituting the Values
Substituting the given values:

$$m = \frac{6.28 \times 10^{-3} \times 0.5}{4\pi \times 10^{-7} \times 5}.$$

Simplifying:

$$m = \frac{6.28 \times 10^{-3} \times 0.5}{12.56 \times 10^{-7}}.$$

Further simplification:

$$m = \frac{3.14 \times 10^{-3}}{12.56 \times 10^{-7}}.$$

Calculating:

$$m = 500.$$

Therefore, the value of $m$ is 500.