Question 285: Magnetism and Matter

Question

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

Accepted Answer

The magnetic field inside a solenoid can be calculated using the formula:
$$B = \mu_0 n I$$
where:
$B$ is the magnetic field in teslas (T),
$\mu_0$ is the permeability of free space ($4\pi 	imes 10^{-7} \mathrm{~Tm/A}$),
$n$ is the number of turns per unit length of the solenoid (turns/m),
$I$ is the current in amperes (A).
Given:
The magnetic field $B = 6.28 	imes 10^{-3} \mathrm{~T}$,
The current $I = 5 \mathrm{~A}$,
The length of the solenoid $L = 0.5 \mathrm{~m}$,
First, let's calculate the number of turns per unit length $n$, which is $n = \frac{m}{L}$ where $m$ is the total number of turns and $L$ is the length of the solenoid.
Rearrange the formula for $B$ to solve for $m$:
$$B = \mu_0 \frac{m}{L} I$$
Therefore,
$$m = \frac{B L}{\mu_0 I}$$
Substituting the values we have:
$$m = \frac{(6.28 	imes 10^{-3} \mathrm{T}) (0.5 \mathrm{m})}{(4\pi 	imes 10^{-7} \mathrm{Tm/A}) (5 \mathrm{A})}$$
$$m = \frac{6.28 	imes 10^{-3} 	imes 0.5}{4\pi 	imes 10^{-7} 	imes 5}$$
$$m = \frac{6.28 	imes 0.5 	imes 10^{-3}}{20\pi 	imes 10^{-7}}$$
$$m = \frac{3.14 	imes 10^{-3}}{20 \pi 	imes 10^{-7}}$$
$$m = \frac{3.14 	imes 10^{-3}}{20 	imes 3.14 	imes 10^{-7}}$$
$$m = \frac{1}{20 	imes 10^{-4}}$$
$$m = \frac{1 	imes 10^4}{20}$$
$$m = 500$$
Therefore, the value of $m$ is 500 turns.

Magnetism and Matter

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

Integer / Numeric Answer Type