A planar loop of wire rotates in a uniform magnetic field. Initially, at t=0, the plane of the loop is perpendicular to the magnetic field. If it rotates with a period of 10s about an axis in its plane then the magnitude of induced emf will be maximum and minimum, respectively at:

Given data: 

$\mathrm{t}=0$and rotates with period of 10 s

Concept used: gauss law for magnetism 

Detailed solution:

Time period,$\mathrm{T}=10 \mathrm{s}$

Axis of rotation $\to y-axis.$

$\overrightarrow{\mathrm{B}}=\mathrm{B}(-\hat{\mathrm{k}})$

at some time t, area vector makes angle $\text{'}\theta \text{'}with \overrightarrow{\mathrm{B}}.flux,\varphi =\overrightarrow{B}·\overrightarrow{A}$

$=\mathrm{BAcos}\theta$

$\mathrm{emf},\epsilon =\left|\frac{-\mathrm{d}\varphi }{\mathrm{dt}}\right|=\left(\mathrm{BAsin}\theta \frac{\mathrm{d}\theta }{\mathrm{dt}}\right)$

$=\mathrm{BA}\omega \sin \theta$

When $\theta =\frac{\pi }{2},\epsilon ={\epsilon }_{\max }$

So,$\theta =n\pi +\frac{\pi }{2} \left( n\to integer\right)$

When $\theta =0,\epsilon ={\epsilon }_{\min }=0$

So,$\theta =n\pi$

$\text{ for }\epsilon ={\epsilon }_{\max },\theta =\omega \mathrm{t}=\mathrm{n}\pi +\frac{\pi }{2}$

$\Rightarrow \mathrm{t}=\frac{\mathrm{n}\pi }{\omega }+\frac{\pi }{2\omega }=\frac{\mathrm{n}\pi }{\left(\frac{2\pi }{\mathrm{T}}\right)}+\frac{\pi }{2\times \left(\frac{2\pi }{\mathrm{T}}\right)}$

$\left.\mathrm{t}=\left(\frac{\mathrm{n}\pi }{2\pi }\right)+\frac{\pi }{\mathrm{T}}\right)$

$=\frac{2\left(\frac{2\pi }{T}\right)}{2}+\frac{\mathrm{T}}{4}=\frac{\mathrm{n}\times 10}{2}+\frac{10}{4}$

$=5\mathrm{n}+2.5$

=$2.5 \mathrm{s},5+2.5,5\times 2+2.5\dots$

$=2.5 \mathrm{s},7.5 \mathrm{s},12.5 \mathrm{s}\dots$

$\text{ for }\epsilon ={\epsilon }_{\min },\theta =\omega \mathrm{t}=\mathrm{n}\pi$

$\Rightarrow \mathrm{t}=\frac{\mathrm{n}\pi }{\omega }=\frac{\mathrm{n}\pi }{\left(\frac{2\pi }{\mathrm{T}}\right)}=\frac{\mathrm{nT}}{2}$

$\Rightarrow \mathrm{t}=\frac{\mathrm{n}\times 10}{2}=5\mathrm{n}$

$\Rightarrow \mathrm{t}=\frac{\mathrm{n}\times 10}{2}=5\mathrm{n}$