A long solenoid of radius R, and having N turns power unit length carries a time dependent current I(t) = I_{0} cos (ωt). The magnitude of induced electric field at a distance R/2 radially from the axis of the solenoid is
(a) \frac{R}{2} \mu_{0} N I_{0} \omega \sin (\omega t )
(b) \frac{R}{4} \mu_{0} N I_{0} \omega \cos (\omega t )
(c) \frac{R}{4} \mu_{0} N I_{0} \omega \cos (\omega t )
(d) R \mu_{0} N I_{0} \omega \sin (\omega t)
We know, H = NI
\begin{aligned}&\text { So, } H=N I_{0} \cos (w t) \\&B=H_{0} H \\&B=\mu_{0} N I_{0} \cos (w t)\end{aligned}According to maxwell second equation,
\begin{aligned}&\oint E \cdot \overrightarrow{d l}=-\int \frac{\overrightarrow{d B}}{d t} \cdot \overrightarrow{d s} \\&\text { So, } E .2 \pi \frac{R}{2}=\mu_{0} N I_{0} \frac{d}{d t}[\cos (w t)] \times \frac{\pi R^{2}}{y} \\&E=\frac{\mu_{0} N I_{0} w R \sin w(t)}{y}\end{aligned}Hence, the correct option is (c).