A conducting circular loop is rotated about its diameter at a constant angular speed of 100 rad/s in a magnetic field of 0.5 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 mV. The radius of the loop is ____ mm. (Take $$\pi = \frac{22}{7}$$)

A circular loop of radius $$r$$ rotates about its diameter in a magnetic field $$B = 0.5$$ T at $$\omega = 100$$ rad/s.

The induced EMF is $$\varepsilon = NBA\omega\sin\theta$$ where $$A = \pi r^2$$, $$N = 1$$.

At $$\theta = 30°$$: $$\varepsilon = B\pi r^2 \omega \sin 30°$$

$$15.4 \times 10^{-3} = 0.5 \times \frac{22}{7} \times r^2 \times 100 \times 0.5$$

$$15.4 \times 10^{-3} = 0.5 \times \frac{22}{7} \times r^2 \times 50$$

$$15.4 \times 10^{-3} = \frac{25 \times 22}{7} \times r^2 = \frac{550}{7} r^2$$

$$r^2 = \frac{15.4 \times 10^{-3} \times 7}{550} = \frac{107.8 \times 10^{-3}}{550} = \frac{0.1078}{550} = 1.96 \times 10^{-4}$$ m$$^2$$

$$r = 0.014$$ m $$= 14$$ mm.

The answer is 14.