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

In a uniform magnetic field of induction $B$, a wire in the form of a semicircle of radius $r$ rotates about the diameter of the circle with an angular frequency $\omega$. The axis of rotation is perpendicular to the field. If the total resistance of the circuit is $R$, the mean power generated per period of rotation is (A) $\frac{(B \pi r \omega)^{2}}{2 R}$ (B) $\frac{\left(B \pi r^{2} \omega\right)^{2}}{8 R}$ (C) $\frac{B \pi r^{2} \omega}{2 R}$ (D) $\frac{\left(B \pi r \omega^{2}\right)^{2}}{8 R}$

In a uniform magnetic field of induction $B$, a wire in the form of a semicircle of radius $r$ rotates about the diameter of the circle with an angular frequency $\omega$. The axis of rotation is perpendicular to the field. If the total resistance of the circuit is $R$, the mean power generated per period of rotation is
(A) $\frac{(B \pi r \omega)^{2}}{2 R}$
(B) $\frac{\left(B \pi r^{2} \omega\right)^{2}}{8 R}$
(C) $\frac{B \pi r^{2} \omega}{2 R}$
(D) $\frac{\left(B \pi r \omega^{2}\right)^{2}}{8 R}$

Key Concepts

Faraday's Law of Electromagnetic Induction

This principle states that a time-varying magnetic flux through a closed circuit induces an electromotive force (EMF) in that circuit. It forms the basis for understanding how mechanical motion in magnetic fields, such as rotation, converts into electrical energy.

Motional EMF

Motional EMF refers to the voltage generated when a conductor moves through a magnetic field. The movement results in charge separation due to the Lorentz force, which manifests as an induced EMF. This concept is essential in analyzing systems where mechanical motion leads to electrical power generation.

Average Power Dissipation in Resistive Circuits

In circuits with resistance, the induced EMF drives a current that dissipates power as heat. The calculation of average power involves determining the instantaneous power (which is proportional to the square of the EMF divided by the resistance) and then averaging it over one complete cycle. This is crucial for assessing the efficiency of energy conversion in AC systems.

Angular Motion in Electromagnetic Systems

The angular velocity of a moving conductor in a magnetic field determines the rate at which the orientation – and thus the magnetic flux – changes over time. This connection between rotational dynamics and electromagnetic induction is pivotal in analyzing systems like generators and AC power sources.

Transcript

00:01 Hello students in this question we have a uniform magnetic induction b a wire in the form of semi circle of radius semi circle semi circle of radius small r is rotating about the diameter of the circle with angular frequency omega okay, axis of rotation is perpendicular to the field so if total resistance is r then we have to determine the mean power generated per period of revolution okay so first of all we can find the emf so that will be given by minus of d5 divided by dt so it can be written as minus of d by d t of b a cos theta okay and this theta can be written as omega modiclar by t because it is rotating with angle of frequency omega so after differentiating this emf will be equals to the b a omega sine omega t okay now we can obtain the induced current i so that will be emf divided the resistance r so this is b omega a divided by resistance r and this sign omega t okay now we can find the power so that will be given by i square multiplied resistance r so i which is this value so b omega a divided by r and this sine is square omega t okay so sine is square omega t okay so sine omega t we can separately write okay and this whole square and multiply resistance r so we will obtain from here that the power p is equals to the power p is equals to the omega square b square a square divided by resistance r and this sine square of omega t okay now we have to determine the mean power so we know that this is a constant value and this has the mean value of 1 by 2 and we can write the area of the area of the semicircle so that will be pi r squared by 2 okay so average power or mean power will be equals to the omega b and this a which is pi r square divided by 2 capital r and this whole square and this 1 by 2 okay so after rearranging this equation we will obtain that mean power is equals to the omega b pi r square and this whole square divided by eight r okay so this is the answer for the problem okay so here we have only square in the numerator known in the denominator okay so this is the answer for the problem okay so from the given options option b is correct okay option b is the answer here...

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