Question

A circular coil of wire consisting of 100 turns, each of radius 8.0 cm carries a current of 0.40 A. What is the magnitude of the magnetic field B at the center of the coil?

Answer 1

Hint: Biot-Savart’s law is used to find the magnetic field due to a current. For a circular coil of radius R and N turns carrying current I, the magnitude of the magnetic field at a point on its axis at a distance x from its center can be calculated by using the equation deduced by using Biot-Savart’s law​. Put $x=0$, to find the magnitude of the magnetic field at the center of the coil.

Formula used:
$B={\displaystyle \frac{{\mu }_{0}I{R}^{2}N}{2{\left(x^2+R^2\right)}^{3/2}}}$
where,
$B$ represents the magnitude of the magnetic field,
$I$ represents the current in the conductor,
$R$ represents the radius of the circular coil,
$N$ represents the number of turns in the coil per unit length,
and $x$ is the distance at which the magnetic field is being calculated.

Complete step by step answer:
Number of turns in a circular coil is given by,
N=100
The radius of each turn is given by,
$R=8.0cm=0.08m$
Current flowing in the coil is given by,
$I=0.4A$
We know that the magnitude of the magnetic field $B$ at the centre of the coil is given by,

$\begin{array}{rl}& B={\displaystyle \frac{{\mu }_{0}NI}{2r}}\\ & B={\displaystyle \frac{{\mu }_0}{4\pi }}{\displaystyle \frac{NI}{r}}\times 2\pi \end{array}$
where,
${\mu }_0$ represents the permeability of free space and is numerically equal to $\text{4}\pi \times \text{1}{\text{0}}^{-7}m/A$,
$N$ represents the number of turns per unit length,
$r$ represents the radius of the circular coil.

$\begin{array}{rl}& B={\displaystyle \frac{4\pi \times {10}^{-7}\times 0.4\times 100\times 2\pi }{4\pi \times 0.08}}\\ & B=3.14\times {10}^{-4}T\end{array}$
Hence, the magnitude of the magnetic field is $B=3.14\times {10}^{4}T$.

Additional Information
A current flowing through a conductor produces a magnetic field around it. The current is nothing but the flow of electrons, thus, indirectly flowing charges or accelerating charges produces a magnetic field. The Tesla (T) is the SI unit of the magnetic field, whereas Gauss (G) is its CGS unit.

Note:
The magnetic field is a vector quantity, therefore, it also has a direction. The direction of the magnetic field is given by the right-hand thumb rule. In the case of a circular loop carrying a current I, the direction of the magnetic field is along the direction of the central axis of the loop. If the circular loop is in the XY plane, the direction of the magnetic field is along the positive z-axis when the current flows in the anticlockwise direction in the loop, and the direction of the magnetic field is along the negative z-axis when the current flows in the clockwise direction in the loop.