Question

Two rings X and Y are placed in such a way that their axes are along the X and the Y axes respectively and their centers are at the origin. Both the rings X and Y have the same radii of 3.14 cm. If the current through X and Y rings are 0.3A and 0.4A respectively, find the value of the resultant magnetic field at the origin.

Answer 1

Hint: Magnetic field at the centre of a ring is given by:
$B = \dfrac{{{\mu _0}I}}{{2r}}$ (I is the current flowing in the circular loop, r is the distance of the loop from the magnetic field, ${\mu _0}$ is the permeability of free space)
Using the above relation we will find the magnetic field at the origin.

Complete step by step solution:
Let's discuss the magnetic field and how it is produced.
The space around a current carrying conductor, in which its magnetic effect can be experienced, is called magnetic field.
The current through the conductor produces something in the space around the conductor, called magnetic field and this magnetic field exerts a force on any magnetic pole placed in the field. Direction of this induced magnetic field lines due to a straight current carrying conductor may be found by applying right hand thumb rule or Maxwell's cork screw rule.
Now, we will calculate the magnetic field at the origin.
$B = \dfrac{{{\mu _0}I}}{{2r}}$(magnetic field given at the centre)
Let ${B_x}$ be the magnetic field due to the ring at the x-axis, ${B_y}$ be the magnetic field due to the ring at the y-axis.
$ \Rightarrow {B_x} = \dfrac{{4\pi \times {{10}^{ - 7}} \times 0.3}}{{2 \times 3.14}} = \dfrac{{3.76 \times {{10}^{ - 7}}}}{{6.28}}$
$ \Rightarrow {B_y} = \dfrac{{4\pi \times {{10}^{ - 7}} \times 0.4}}{{2 \times 3.14}} = \dfrac{{5.0265 \times {{10}^{ - 7}}}}{{6.28}}$
On adding the two magnetic fields
We have;
$ \Rightarrow B = {B_x} + {B_y} $
$\Rightarrow B = \dfrac{{(3.76 + 5.0265) \times {{10}^{ - 7}}}}{{6.28}} $
(We have taken the common term out)
$ \Rightarrow B = \dfrac{{8.78 \times {{10}^{ - 7}}}}{{6.28}}$
$ \Rightarrow B = 1.4 \times {10^{ - 7}}Tesla$

Magnetic field at the origin is $1.4 \times 10^{-7}$ Tesla.

Note: Magnetic field has many applications which we observe in our day to day life such as electromagnets, AC and DC electric motors, electric generators, electric transformers, electromagnetic wave propagation, magnetic levitation and magnetic resonance imaging system, mass spectrometers, gas chromatography etc.