Question

Two circular coils of radii 5cm and 10cm carry currents of 2A. The coils have 50 and 100 turns respectively and are placed in such a way that their planes as well as their centres coincide. Magnitude of magnetic field at the common centre of the coil is
(This question has multiple correct options)
A) $8\pi \times {{10}^{-4}}T$ if currents in the coils are in the same sense
B) $4\pi \times {{10}^{-4}}T$ if currents in the coils are in the opposite sense
C) Zero if currents in the coils are in the opposite sense
D) $8\pi \times {{10}^{-4}}T$ if currents in the coils are in the opposite sense

Answer 1

Hint: This question can be solved by finding the magnitude of the magnetic field due to each current carrying coil at its centre and summing them up properly keeping in mind the direction. If the currents in the coil are in the same sense, the individual magnetic fields due to the two coils will be added and if they are in the opposite sense, they will have to be subtracted.

Formula used:
The magnetic field $B$ at the centre of a coil of $N$ turns and radius $R$ carrying a current $I$
$B=\dfrac{{{\mu }_{0}}NI}{2R}$
where ${{\mu }_{0}}=4\pi \times {{10}^{-7}}m.kg.{{s}^{-2}}{{A}^{-2}}$.

Complete step by step answer:
We will find the magnetic field due to each current carrying coil at its centre. Then depending upon whether we have considered the currents in the coils to be in the same or opposite sense, we will add or subtract the individual magnetic fields respectively to get the total magnetic field at the centre.
The magnetic field $B$ at the centre of a coil of $N$ turns and radius $R$ carrying a current $I$
$B=\dfrac{{{\mu }_{0}}NI}{2R}$ --(1)
where ${{\mu }_{0}}=4\pi \times {{10}^{-7}}m.kg.{{s}^{-2}}{{A}^{-2}}$ .
Let us analyze the question.
Let the current in coil 1 be ${{I}_{1}}=2A$.
The number of turns in coil 1 be ${{N}_{1}}=50$.
The radius of coil 1 is ${{R}_{1}}=5cm=0.05m$ $\left( \because 1cm=0.01m \right)$.
The magnetic field produced at the center due to this coil will be ${{B}_{1}}$.
Let the current in coil 2 be ${{I}_{2}}=2A$.
The number of turns in coil 2 be ${{N}_{2}}=100$.
The radius of coil 2 is ${{R}_{2}}=10cm=0.1m$ $\left( \because 1cm=0.01m \right)$.
The magnetic field produced at the center due to this coil will be ${{B}_{2}}$.
Now using (1), we get,
${{B}_{1}}=\dfrac{{{\mu }_{0}}{{N}_{1}}{{I}_{1}}}{2{{R}_{1}}}=\dfrac{{{\mu }_{0}}\times 50\times 2}{2\times 0.05}=1000{{\mu }_{0}}$ --(2)
Now using (1), we get,
${{B}_{2}}=\dfrac{{{\mu }_{0}}{{N}_{2}}{{I}_{2}}}{2{{R}_{2}}}=\dfrac{{{\mu }_{0}}\times 100\times 2}{2\times 0.1}=1000{{\mu }_{0}}$ --(3)
Case 1) The currents in the coils are in the same sense.
If the currents in the coils are in the same sense, the magnetic fields produced by the individual coils will be in the same direction and hence they will add up. Hence, the total magnetic field $B$ at the centre will be given by
$B={{B}_{1}}+{{B}_{2}}$
Using (2) and (3), we get,
$B=1000{{\mu }_{0}}+1000{{\mu }_{0}}=2000{{\mu }_{0}}=2000\times 4\pi \times {{10}^{-7}}=8\pi \times {{10}^{-4}}T$ $\left( \because {{\mu }_{0}}=4\pi \times {{10}^{-7}}m.kg.{{s}^{-2}}{{A}^{-2}} \right)$
Hence, the total magnetic field at the centre when the currents in the coils are in the same sense is $8\pi \times {{10}^{-5}}T$.
Case 2) When the currents in the coils are in the opposite sense.
When the currents in the coils are in the opposite sense, the magnetic fields produced by them will be in the opposite direction to each other and hence will try to cancel each out. Hence, the total magnetic field $B$ at the centre due to the two coils will be
$B=\left| {{B}_{1}}-{{B}_{2}} \right|$
Using (2) and (3), we get,
$B=\left| 1000{{\mu }_{0}}-1000{{\mu }_{0}} \right|=\left| 0 \right|=0$
Hence, the magnetic fields will cancel each other out completely and hence, the total magnetic field at the centre of the coil will be zero.

Hence, the correct options are A) and C).

Note: Students sometimes do not consider both the cases. It often does not click in the mind of the students that the currents in the coils might be opposite in sense and hence produce opposing magnetic fields. They only consider that the currents in the coils must be in the same sense and add the individual magnetic fields produced by the coils. Students should always consider both the cases as the complete picture cannot be seen if both the cases are not considered individually.