A coil in the form of a quarter circle rotates in the OXY plane and about OZ axis with a constant angular velocity ω. A constant magnetic field -B0k^ exists in the region x≥0 . The induced emf E around the coil is plotted versus angle turned θ. Consider the anti clockwise emf positive. The position θ=0 is shown in the figure. The correct graph is

ϕ=12a2θB0E=dϕdt=B0a22dθdt=B0a2ω20≤θ≤π2, The value is constant E anticlockwise In the range π2≤θ≤πϕ=12a2π2B0,dϕdt=0π≤θ≤3π2,dϕdt=−B02ω2, Constant negative value 3π2≤θ≤2π,dϕdt=0

A coil in the form of a quarter circle rotates in the OXY plane and about OZ axis with a constant angular velocity ω. A constant magnetic field -B0k^ exists in the region x≥0 . The induced emf E around the coil is plotted versus angle turned θ. Consider the anti clockwise emf positive. The position θ=0 is shown in the figure. The correct graph is

ϕ=12a2θB0E=dϕdt=B0a22dθdt=B0a2ω20≤θ≤π2, The value is constant E anticlockwise In the range π2≤θ≤πϕ=12a2π2B0,dϕdt=0π≤θ≤3π2,dϕdt=−B02ω2, Constant negative value 3π2≤θ≤2π,dϕdt=0

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A coil in the form of a quarter circle rotates in the OXY plane and about OZ axis with a constant angular velocity $ω$ . A constant magnetic field ${-B}_{0}$ $\hat{k}$ exists in the region $x \geq 0$ . The induced emf E around the coil is plotted versus angle turned $θ$ . Consider the anti clockwise emf positive. The position $θ = 0$ is shown in the figure. The correct graph is

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Detailed Solution

$ϕ = (\frac{1}{2} a^{2} θ) B_{0}$
$E = \frac{dϕ}{dt} = \frac{B_{0} a^{2}}{2} \frac{dθ}{dt} = \frac{B_{0} a^{2} ω}{2} 0 \leq θ \leq \frac{π}{2}$ , The value is constant E anticlockwise In the range $\frac{π}{2} \leq θ \leq πϕ = \frac{1}{2} a^{2} (\frac{π}{2}) B_{0}, \frac{dϕ}{dt} = 0$
$π \leq θ \leq \frac{3 π}{2}, \frac{dϕ}{dt} = - \frac{B_{0}^{2} ω}{2}$ , Constant negative value $\frac{3 π}{2} \leq θ \leq 2 π, \frac{dϕ}{dt} = 0$