Question

Two identical circular coils each of radius \(R\) and number of turns \(N\) are arranged coaxially with a separation of \(R\) between their centers. If the current through each coil is \(I\), then the maximum resultant magnetic field induced at the midpoint of the line joining the centers of the two coils is

• A. \(\dfrac{2\mu_0NI}{R}\)
• B. \(\dfrac{4\mu_0NI}{5R}\)
• C. \(\dfrac{8\mu_0NI}{5\sqrt5\,R}\)
• D. \(\dfrac{\mu_0NI}{\pi R}\)

Hint:

The magnetic field on the axis of a circular coil is \[ \boxed{ B = \frac{\mu_0NI R^2} {2(R^2+x^2)^{3/2}}. } \] At the midpoint between two identical coaxial coils (Helmholtz arrangement), the magnetic fields add in the same direction.

Answer 1

The Correct Option is C

Step 1: Find the magnetic field due to one coil. The magnetic field on the axis of a circular coil at a distance \(x\) from its centre is \[ B = \frac{\mu_0NI R^2} {2(R^2+x^2)^{3/2}}. \] Here, \[ x=\frac{R}{2}. \] Hence, \[ B_1 = \frac{\mu_0NI R^2} {2\left(R^2+\dfrac{R^2}{4}\right)^{3/2}}. \] Since \[ R^2+\frac{R^2}{4} = \frac{5R^2}{4}, \] \[ B_1 = \frac{\mu_0NI R^2} {2\left(\dfrac{5R^2}{4}\right)^{3/2}} = \frac{4\mu_0NI}{5\sqrt5\,R}. \]

Step 2: Find the resultant magnetic field. The fields due to the two identical coils are in the same direction at the midpoint. Therefore, \[ B = 2B_1 = 2\left(\frac{4\mu_0NI}{5\sqrt5\,R}\right). \] Hence, \[ B = \frac{8\mu_0NI}{5\sqrt5\,R}. \]

Step 3: Write the answer. Therefore, \[ \boxed{ \frac{8\mu_0NI}{5\sqrt5\,R} }. \] Thus, \[ \boxed{(C)} \] is the correct answer.