Question

A wire carrying a current of $4\,{\text{A}}$ is in the form of a circle. It is necessary to have a magnetic field of induction ${10^{ - 6}}{\text{T}}$ at the centre. The radius is:
A. $2.51\,{\text{m}}$
B. $25.1\,{\text{m}}$
C. $251\,{\text{m}}$
D. $0.251\,{\text{m}}$

Answer 1

First of all, we will find the expression of the magnetic field and substitute the required values in it. We will manipulate the expression accordingly and obtain the result.

Complete step by step answer:
In the given problem, we are supplied with the following data:
The amount of current that the wire is carrying is $4\,{\text{A}}$ .
The required magnetic field at the centre is ${10^{ - 6}}{\text{T}}$ .
The wire forms a circle.
We are required to form the radius of the wire loop.

For this we apply the formula which gives the magnetic field.
$B = \dfrac{{{\mu _0}i}}{{2r}}$ …… (1)
Where,
$B$ indicates magnetic field.
${\mu _0}$ indicates permeability constant.
$i$ indicates the amount of current flowing through the wire.
$r$ indicates the radius of the wire loop.

Now, substituting the required values in the equation (1), we get:

$$B = \dfrac{{{\mu _0}i}}{{2r}} \\\ \Rightarrow {10^{ - 6}} = \dfrac{{4\pi \times 1{0^{ - 7}} \times 4}}{{2r}} \\\ \Rightarrow r = \dfrac{{4\pi \times 1{0^{ - 7}} \times 4}}{{{{10}^{ - 6}}}} \\\ \Rightarrow r = 2.512\,{\text{m}} \\\$$

$r \sim 2.51\,{\text{m}}$
Hence, the radius of the wire loop is $2.51\,{\text{m}}$ .
The correct option A.

Additional information:
A magnetic field is a vector field that determines the magnetic effect on electrical charges, electrical currents, and magnetised objects that pass. A force perpendicular to its own velocity and to the magnetic field is encountered by a charge travelling in a magnetic field.
In permanent magnets, which draw on magnetic materials such as iron and attract or repel other magnets, the effects of magnetic fields are frequently seen. Besides that, by affecting the motion of their outer atomic electrons, a magnetic field that varies with position can exert a force on a variety of non-magnetic materials.

Note: While solving this problem, we need to know about the magnetic field and its effects. The magnetic field depends on the current flowing through the wire, number of turns and material used.