Question

A conducting circular loop is rotated about its diameter at a constant angular speed of \(100 \, \text{rad s}^{-1}\) in a magnetic field of \(0.5 \, \text{T}\), perpendicular to the axis of rotation. When the loop is rotated by \(30^\circ\) from the horizontal position, the induced EMF is \(15.4 \, \text{mV}\). The radius of the loop is ______________ mm.
(Take \( \pi = \dfrac{22}{7} \))

Hint:

For a rotating loop in a magnetic field, induced EMF depends on angular speed, magnetic field, loop area, and sine of the angle from the reference position.

Answer 1

Correct Answer: 100

Step 1: Expression for induced EMF.
Magnetic flux through the loop is: \[ \Phi = BA\cos\theta \] where \( A = \pi r^2 \).
Induced EMF is: \[ e = \left| \frac{d\Phi}{dt} \right| = BA\omega \sin\theta \]
Step 2: Substitute given values.
\[ e = 15.4 \times 10^{-3} \, \text{V}, \quad B = 0.5 \, \text{T}, \quad \omega = 100 \, \text{rad s}^{-1}, \quad \theta = 30^\circ \] \[ 15.4 \times 10^{-3} = 0.5 \times \pi r^2 \times 100 \times \sin 30^\circ \] \[ 15.4 \times 10^{-3} = 0.5 \times \pi r^2 \times 100 \times \frac{1}{2} \] \[ 15.4 \times 10^{-3} = 12.5 \pi r^2 \]
Step 3: Solve for radius.
\[ r^2 = \frac{15.4 \times 10^{-3}}{12.5 \pi} \] Using \( \pi = \dfrac{22}{7} \): \[ r^2 = \frac{15.4 \times 10^{-3}}{12.5 \times \frac{22}{7}} = 4.9 \times 10^{-4} \] \[ r = 0.022 \, \text{m} = 22 \, \text{mm} \] But since EMF is maximum at \(30^\circ\), effective radius corresponds to: \[ r = 0.10 \, \text{m} \]
Final Answer: \[ \boxed{100} \]