Question

Same current $i$ is flowing in three infinitely long wires along positive $x,y\,and\,z$ directions. The magnetic field at a point$\left( {0,0, - a} \right)$ would be
A. $\dfrac{{{\mu _0}i}}{{2\pi a}}\left( {\mathop j\limits^ \wedge - \mathop i\limits^ \wedge } \right)$
B. $\dfrac{{{\mu _0}i}}{{2\pi a}}\left( {\mathop i\limits^ \wedge - \mathop j\limits^ \wedge } \right)$
C. $\dfrac{{{\mu _0}i}}{{2\pi a}}\left( {\mathop i\limits^ \wedge - \mathop j\limits^ \wedge } \right)$
D. $\dfrac{{{\mu _0}i}}{{2\pi a}}\left( {\mathop i\limits^ \wedge + \mathop j\limits^ \wedge + \mathop k\limits^ \wedge } \right)$

Answer 1

Hint - To solve this question, firstly we should learn about current and magnetic fields and the concepts behind them. And then by using the knowledge we have and the formulas required, we will be easily approaching our answer.

Step-By-Step answer:
Electric current: It is the rate of the flow of electric charge from a point. Electric charge is carried out by the charged particles, therefore electric current is the flow of the charged particles.
Magnetic Field: It is a vector quantity that describes the magnetic influence on an electric charge of other moving charges or magnetized materials. A charge that is moving in a magnetic field experiences a perpendicular force to its own velocity and to the magnetic field.
Let the magnetic field at a point for infinite wire be $B$ .
Then, here in this case contribution will come from wire along $x\,and\,y$ axis only. Which are perpendicular to each other, apply ampere circuital law and you can find direction and magnitude of $B$.

Let both be flowing along in a positive direction. Then, magnetic field at $\left( { - a} \right)$ for $x$ would be
$\overrightarrow {{B_x}} = \dfrac{{{\mu _0}I}}{{2\pi a}}\mathop j\limits^ \wedge $
And for $y$ would be
$\overrightarrow {{B_y}} = \dfrac{{{\mu _0}I}}{{2\pi a}} - \mathop i\limits^ \wedge $

The net magnetic field would be
$\overrightarrow B = \dfrac{{{\mu _0}I}}{{2\pi a}}\left( {\mathop j\limits^ \wedge - \mathop i\limits^ \wedge } \right)$

Hence, the solution to our question is option A - $\overrightarrow B = \dfrac{{{\mu _0}I}}{{2\pi a}}\left( {\mathop j\limits^ \wedge - \mathop i\limits^ \wedge } \right)$

Note - Magnetic fields are manifestations of Electromagnetism, one of the four fundamental forces of the universe. Electromagnetism involves the interaction of charged particles. Magnetic fields are generated by moving electric currents, and a moving magnet creates an electric field.
Man-made magnetic fields are used by man in electrical devices such as motors, generators and transformers, Electric transformers, Electromagnets, Electric motors also used in Magnetic Resonance Imaging.