Question

A part of a long wire A carrying a current I is bent into a circle of radius r as shown in the figure. The net magnetic field at the centre O of the circular loop is

\[\begin{align}
  & A\,\dfrac{{{\mu }_{0}}i}{4r} \\
 & B\,\dfrac{{{\mu }_{0}}i}{2r} \\
 & C\,\dfrac{{{\mu }_{0}}i}{2\pi r}(\pi +1) \\
 & D\,\dfrac{{{\mu }_{0}}i}{2\pi r}(\pi -1) \\
\end{align}\]

Answer 1

Hint: The net magnetic field at the centre O of the circular loop is the sum of the magnetic field due to the circular part of the wire and the magnetic field due to the straight part of the wire. So, we will add the magnetic fields of the circular part and the straight part of the wires.

Formula used:
\[B=\dfrac{{{\mu }_{0}}i}{2\pi r}\]
\[B=\dfrac{{{\mu }_{0}}i}{2r}\]

Complete step by step solution:
At the centre O of the circular loop, the magnetic field due to the straight part of the wire is given as follows.
\[{{B}_{1}}=\dfrac{{{\mu }_{0}}i}{2\pi r}\]
In this case, ‘r’ represents the distance between the centre O and the straight wire.
At the centre O of the circular loop, the magnetic field due to the circular part of the wire is given as follows.
\[{{B}_{2}}=\dfrac{{{\mu }_{0}}i}{2r}\]
In this case, ‘r’ represents the radius of the circular loop.
Therefore, the net magnetic field at the centre O of the circular loop is the sum of the magnetic field due to the circular part of the wire and the magnetic field due to the straight part of the wire.
Thus, the net magnetic field at the centre O of the circular loop is,
\[B={{B}_{1}}+{{B}_{2}}\]
Now substitute the expressions of the magnetic fields in the above equation.
\[\begin{align}
  & B=\dfrac{{{\mu }_{0}}i}{2\pi r}+\dfrac{{{\mu }_{0}}i}{2r} \\
 & \Rightarrow B=\dfrac{{{\mu }_{0}}i}{2r}\left( \dfrac{1}{\pi }+1 \right) \\
 & \Rightarrow B=\dfrac{{{\mu }_{0}}i}{2r}\left( \dfrac{1+\pi }{\pi } \right) \\
 & \therefore B=\dfrac{{{\mu }_{0}}i}{2\pi r}(1+\pi ) \\
\end{align}\]
\[\therefore \]The value of the net magnetic field at the centre O of the circular loop is\[\dfrac{{{\mu }_{0}}i}{2\pi r}(1+\pi )\], thus, the option (C) is correct.

Note: The expressions for the magnetic fields along the straight wire and the circular loop will be different. In order to solve this problem, the formulae for computing the magnetic field along the straight and the circular parts should be known.