Question

What should be the current in a circular coil of radius $5\,cm$ to annule ${B_H} = 5 \times {10^{ - 5}}\,T$ ?
A. $0.4\,A$
B. $4\,A$
C. $40\,A$
D. $1\,A$

Answer 1

To get the value of current for a circular whose radius is given in centimetres, which we will convert to metres, and the earth's horizontal component is also supplied. As a result, we'll use the magnetic field formula to arrive at our desired result.

Complete step by step answer:
In the question, we are provided with some terms; i.e. The radius of the circular loop,
$\left( r \right) = 5cm = 5 \times {10^{ - 2}}m$
And the horizontal component of the earth $\left( {{B_H}} \right) = 5 \times {10^{ - 5}}T$ i.e. we can also say that this is the horizontal component of the magnetic. So, now, the magnetic field $B$ at the centre of a circular ring of radius $r$ carrying a current $I$ may be calculated using the Biot-Savart law.
$B = \dfrac{{{\mu _0}I}}{{2r}}$
where ${\mu _0}$ is magnetic permeability of vacuum and $= 4\pi \times {10^{ - 7}}H{m^{ - 1}}$
${B_H} = \dfrac{{{\mu _0}I}}{{2r}} \\\ \Rightarrow I = \dfrac{{2R{B_H}}}{{{\mu _0}}} \\\$
We acquire what we want by putting in the specified values.
$I = \dfrac{{2 \times 5 \times {{10}^{ - 2}} \times 3 \times {{10}^{ - 5}}}}{{4\pi \times {{10}^{ - 7}}}} \\\ \Rightarrow I = \dfrac{{50}}{{4\pi }} \\\ \Rightarrow I = \dfrac{{50}}{{4 \times 3.14}} \\\ \Rightarrow I = \dfrac{{50}}{{12.56}} \\\ \therefore I = 4A$
Therefore, the current in a circular coil is $4A$.

So, the correct option is B.

Note: The Biot Savart Law is an equation that describes how a continuous electric current generates a magnetic field. It connects the magnetic field to the electric current's magnitude, direction, length, and proximity. Both Ampere's circuital law and Gauss' theorem are consistent with Biot–Savart law. The Biot-Savart law is a fundamental law in magnetostatics, comparable to Coulomb's law in electrostatics.