Question

Two wires of same length are bent to form a circle and a circular loop. Then the ratio of their magnetic fields at the centre will be written as,
$\begin{align}
  & A.{{N}^{2}} \\
 & B.\dfrac{1}{N} \\
 & C.N \\
 & D.\dfrac{1}{{{N}^{2}}} \\
\end{align}$

Answer 1

Hint: As the wire is bent into the form of a circle and circular loop, the circumference will be the same as the length of the wire. Circumference can be found by taking the product of $2\pi $ and the radius of the loop. First of all find the magnetic field of a circle. Then find the magnetic field of the circular loop. Take the ratio between them and arrive at the answer. This will help you in answering this question.

Complete step by step answer:
Let us assume that the wire is having a length of $L$. It is bent in the form of a circle and circular loop. Therefore the circumference of the loop will be the same as the length of the wire. This can be written as,
$L=2\pi r$
Where $r$ be the radius of the loop. The equation can be rearranged as,
$r=\dfrac{L}{2\pi }$
The magnetic field when the wire is bent into a circle can be written as,
${{B}_{1}}=\dfrac{{{\mu }_{0}}I\pi }{4L}$
Here $I$be the current through the circle.
When the wire is bent to form a circular loop, we can write that,
$N\times 2\pi {{r}_{1}}=2\pi r$
Therefore the radius will become,
${{r}_{1}}=\dfrac{r}{N}$
Where $N$ be the number of turns,
Therefore the magnetic field when the wire is bent in the form of circular loop is given as,
${{B}_{2}}=\dfrac{N{{\mu }_{0}}I}{2{{r}_{1}}}=\dfrac{{{N}^{2}}{{\mu }_{0}}I}{2r}=\dfrac{{{N}^{2}}{{\mu }_{0}}\pi I}{4L}$
Taking the ratio between this will give,
\[\dfrac{{{B}_{1}}}{{{B}_{2}}}=\dfrac{\dfrac{{{\mu }_{0}}I\pi }{4L}}{\dfrac{{{N}^{2}}{{\mu }_{0}}I\pi }{4L}}=\dfrac{1}{{{N}^{2}}}\]

Therefore the answer has been obtained as option D.

Note:
In the case of a circle, which is a two dimensional shape, the number of turns will be only one. Therefore in the equation for the circle, the number of turns will be unity. While the circular loop may have more than one turns. Therefore in this case only the number of turn’s term is valid.