A uniform conducting ring of mass m = 2kg, radius r =2m and the resistance 8Ω is kept on smooth, horizontal surface with its plane horizontal. A time varying magnetic field B→=i^+t2j^ Tesla is present in the region, where t is time in second and take upward vertical as y- axis . (Take π2=10 and g=10m/s2), then

Sol: ϕ=BA=t2πr2 ε=dϕdt=2 πr2t i=εR=2πr2tR For toppling τ=μ→×B→≥ mgr iπr21≥ mgr 2πr2tRπr2≥ mgr t≥mgR2π2r3=1sec Heat = ∫i2Rdt=∫2πr2tR2dt.R Heat = 4π2r4R∫01t2dt =803 Joule

A uniform conducting ring of mass m = 2kg, radius r =2m and the resistance 8Ω is kept on smooth, horizontal surface with its plane horizontal. A time varying magnetic field B→=i^+t2j^ Tesla is present in the region, where t is time in second and take upward vertical as y- axis . (Take π2=10 and g=10m/s2), then

Sol: ϕ=BA=t2πr2 ε=dϕdt=2 πr2t i=εR=2πr2tR For toppling τ=μ→×B→≥ mgr iπr21≥ mgr 2πr2tRπr2≥ mgr t≥mgR2π2r3=1sec Heat = ∫i2Rdt=∫2πr2tR2dt.R Heat = 4π2r4R∫01t2dt =803 Joule

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A uniform conducting ring of mass m = 2kg, radius r =2m and the resistance $8 Ω$ is kept on smooth, horizontal surface with its plane horizontal. A time varying magnetic field $\vec{B} = (\hat{i} + t^{2} \hat{j})$ Tesla is present in the region, where t is time in second and take upward vertical as y- axis . (Take $π^{2} = 10 a n d g = 10 m / s^{2}$ ), then

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he ring starts toppling after 3/4 sec

Heat generated through the ring till the instant when the ring starts rotating is $\frac{80}{3}$ Joule

The ring starts toppling after 1 sec

Heat generated through the ring till the instant when the ring starts rotating is zero Joule

answer is A, D.

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Detailed Solution

Sol: $ϕ = B A = (t^{2}) (π r^{2}) ε = \frac{d ϕ}{d t} = 2 π r^{2} t i = \frac{ε}{R} = \frac{2 π r^{2} t}{R}$
For toppling $τ = |\vec{μ} \times \vec{B}| \geq$ mgr
   $(i π r^{2}) 1 \geq m g r \frac{2 π r^{2} t}{R} π r^{2} \geq m g r t \geq \frac{m g R}{2 π^{2} r^{3}} = 1 \sec$
Heat = $\int i^{2} R d t = \int {(\frac{2 π r^{2} t}{R})}^{2} d t . R$
Heat = $\frac{4 π^{2} r^{4}}{R} \int_{0}^{1} t^{2} d t = \frac{80}{3}$ Joule