(C) The induced $emf$ across a rotating rod in a magnetic field is given by $\epsilon = \frac{1}{2} B \omega R^2$.Here,$B = H_E = 0.4 \; G = 0.4 \time

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(C) The induced $emf$ across a rotating rod in a magnetic field is given by $\epsilon = \frac{1}{2} B \omega R^2$.Here,$B = H_E = 0.4 \; G = 0.4 \times 10^{-4} \; T$.The angular velocity $\omega = 2 \pi \nu$,where $\nu = 120 \; rev/min = \frac{120}{60} \; rev/s = 2 \; rev/s$.So,$\omega = 2 \pi \times 2 = 4 \pi \; rad/s$.The length of the spoke $R = 0.5 \; m$.Substituting the values: $\epsilon = \frac{1}{2} \times (0.4 \times 10^{-4}) \times (4 \pi) \times (0.5)^2$.$\epsilon = 0.2 \times 10^{-4} \times 4 \pi \times 0.25 = 0.2 \times 10^{-4} \times \pi = 0.2 \times 3.14159 \times 10^{-4} \approx 6.28 \times 10^{-5} \; V$.The number of spokes is immaterial because the $emf$s across the spokes are in parallel.

Class 12 Physics Electromagnetic Induction Motional EMI (Induced Parameter)

Difficult

$A$ wheel with $10$ metallic spokes,each $0.5 \; m$ long,is rotated with a speed of $120 \; rev/min$ in a plane normal to the horizontal component of the Earth's magnetic field $H_{E}$ at a place. If $H_{E} = 0.4 \; G$ at the place,what is the induced $emf$ between the axle and the rim of the wheel? $(1 \; G = 10^{-4} \; T)$

A
$1.56 \times 10^{-4} \; V$
B
$9.42 \times 10^{-5} \; V$
C
$6.28 \times 10^{-5} \; V$
D
$3.14 \times 10^{-5} \; V$

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Electromagnetic Induction

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Motional EMI (Induced Parameter)

$A$ rectangular wire loop of sides $8 \;cm$ and $2 \;cm$ with a small cut is moving out of a region of uniform magnetic field of magnitude $0.3 \;T$ directed normal to the loop. What is the emf developed across the cut if the velocity of the loop is $1 \;cm \,s^{-1}$ in a direction normal to the $(a)$ longer side,$(b)$ shorter side of the loop? For how long does the induced voltage last in each case?

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$A$ rod of length $2 \, m$ slides with a speed of $5 \, ms^{-1}$ on a rectangular conducting frame as shown in the figure. There exists a uniform magnetic field of $0.04 \, T$ perpendicular to the plane of the figure. If the resistance of the rod is $3 \, \Omega$, the current through the rod is

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$A$ conducting wire of length $2500 \ m$ is kept in the east-west direction at a height of $10 \ m$ from the ground. If it falls freely to the ground,the induced current in the wire is (Resistance of wire $= 25 \sqrt{2} \ \Omega$,acceleration due to gravity $g = 10 \ m/s^2$,$B_H = 2 \times 10^{-5} \ T$). (in $A$)

DifficultMHT CET 2023

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$A$ circular coil of radius $30 \ cm$,having $1$ turn and a resistance of ${\pi ^2} \ \Omega$,is placed in a magnetic field of $10^{-2} \ T$. The coil rotates about its diameter with a speed of $200 \ rpm$ in a direction perpendicular to the magnetic field. The value of the induced $AC$ current in the coil will be . . . . . . $mA$.

Medium

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$A$ bicycle wheel of radius $0.4\, m$ has $20$ spokes. It is rotating at the rate of $180$ revolutions per minute, perpendicular to the horizontal component of the Earth's magnetic field of $0.4 \times 10^{-4}\, T$. The $emf$ induced between the rim and the centre of the wheel will be

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$A$ rod of length $2 \, m$ slides with a speed of $5 \, ms^{-1}$ on a rectangular conducting frame as shown in the figure. There exists a uniform magnetic field of $0.04 \, T$ perpendicular to the plane of the figure. If the resistance of the rod is $3 \, \Omega$, the current through the rod is

$A$ bicycle wheel of radius $0.4\, m$ has $20$ spokes. It is rotating at the rate of $180$ revolutions per minute, perpendicular to the horizontal component of the Earth's magnetic field of $0.4 \times 10^{-4}\, T$. The $emf$ induced between the rim and the centre of the wheel will be

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