Question

In the given figure, what is the magnetic field at the point $'O'$?

A.$\dfrac{{{\mu }_{0}}I}{4\pi r}+\dfrac{{{\mu }_{0}}I}{2\pi r}$
B.$\dfrac{{{\mu }_{0}}I}{4\pi r}$
C.$\dfrac{{{\mu }_{0}}I}{4r}+\dfrac{{{\mu }_{0}}I}{4\pi r}$
D.$\dfrac{{{\mu }_{0}}I}{4r}-\dfrac{{{\mu }_{0}}I}{4\pi r}$

Answer 1

Hint: The simplest way to solve this question would be to divide the wire in three parts and then calculate the magnetic field due to each part separately and finally add them up. The three parts in which we have divided the wire are: semi-circular parts and the two straight parts.

Complete answer:
First let us find the magnetic field at point $O$ due to the semi-circular portion. Magnetic field at the center of a complete circle of radius $r$ and current $I$ is given by:
${{B}_{circ}}=\dfrac{{{\mu }_{0}}I}{2r}$
For semi-circle, the magnetic field will be:
$\begin{align}
  & {{B}_{semi-circ}}=\dfrac{\dfrac{{{\mu }_{0}}I}{2r}}{2} \\
 & \Rightarrow {{B}_{semi-circ}}=\dfrac{{{\mu }_{0}}I}{4r} \\
\end{align}$
By right hand thumb rule, we can conclude that the direction of the magnetic field is outward to the plane of the figure.
Now, let us calculate the magnetic field for the straight wires:
The magnetic field due to the vertical wire will be zero, while for the horizontal wire it will be:
${{B}_{hori}}=\dfrac{{{\mu }_{0}}I}{4\pi r}$
By right hand thumb rule, we can conclude that the direction of the magnetic field is outward to the plane of the figure.
Since the direction of magnetic field of both the semi-circular wire and the horizontal wire is the same, i.e., outwards to the plane of the figure, hence they will get added up to find the total magnetic field.
Hence the total magnetic field at point $O$ will be:
$\begin{align}
  & {{B}_{total}}={{B}_{semi-circ}}+{{B}_{hori}} \\
 & \therefore {{B}_{total}}=\dfrac{{{\mu }_{0}}I}{4r}+\dfrac{{{\mu }_{0}}I}{4\pi r} \\
\end{align}$

So,Option C is the correct answer.

Note:
Finding the direction of the magnetic field of each part is really important otherwise we would get wrong results. Here, the direction of the magnetic field in both the semi-circular and horizontal wires was the same, therefore we added them up, if it was different then the answer would have been different.