In a uniform magnetic field of induction B , a wire, in the form of...

Magnetic flux linked

= \text{BA} \text\cos \omega t = \dfrac{\text{B} \pi r^{2} \text\cos \omega t}{2}

$∴$ Induced emf \text{e} = \dfrac{- \text{d} ϕ}{\text{dt}} = \dfrac{- 1}{2} \text{B} \pi r^{2} \omega \text\sin \omega t

$∴$ Power $= \dfrac{e^{2}}{\text{R}} = \dfrac{B^{2} {π}^{2} r^{4} {ω}^{2} \sin^{2} \omega t}{4 \text{R}}$

& & & & & & & & $= \dfrac{{\left(\text{B} \pi r^{2} \omega\right)}^{2}}{4 \text{R}} \sin^{2} \omega t$

$∵< \sin^{2} \omega t > = 1 / 2$

$∴$ Mean power generated
& & & & & & & & $= \dfrac{{\left(\text{B} \pi r^{2} \omega\right)}^{2}}{4 \text{R}} \times \dfrac{1}{2} = \dfrac{{\left(\text{B} \pi r^{2} \omega\right)}^{2}}{8 \text{R}}$.