A circular loop of area A is placed perpendicular to a...

A circular loop of area $A$ is placed perpendicular to a time-varying magnetic field of magnitude $B(t)=B_{0}+a t+b t^{2},$ where $B_{0}, a,$ and $b$ are constants. a) What is the magnetic flux through the loop at $t=0 ?$ b) Derive an equation for the induced potential difference in the loop as a function of time. c) What are the magnitude and the direction of the induced current if the resistance of the loop is $R ?$

A circular loop of area $A$ is placed perpendicular to a time-varying magnetic field of magnitude $B(t)=B_{0}+a t+b t^{2},$ where $B_{0}, a,$ and $b$ are constants.
a) What is the magnetic flux through the loop at $t=0 ?$
b) Derive an equation for the induced potential difference in the loop as a function of time.
c) What are the magnitude and the direction of the induced current if the resistance of the loop is $R ?$

Key Concepts

Induced Current and Ohm's Law

When an emf is induced in a circuit, the resulting current is determined by Ohm's law, which relates the current to the applied voltage (or emf) and the resistance of the circuit. The induced current is calculated by dividing the induced emf by the resistance of the loop. This concept links the phenomena of electromagnetic induction to the practical behavior of electric circuits.

Magnetic Flux

Magnetic flux is the measure of the magnetic field passing through a given surface area. It is defined as the integral of the magnetic field over that area, and for a uniform field perpendicular to the surface it simplifies to the product of the magnetic field magnitude and the area. This concept is central to understanding how changes in magnetic fields interact with physical circuits.

Faraday's Law of Electromagnetic Induction

Faraday's law states that a change in magnetic flux through a circuit induces an electromotive force (emf) in the circuit. The magnitude of this induced emf is equal to the rate at which the flux changes over time. This principle is used to derive the relationship between time-varying magnetic fields and induced potential differences in circuits.

Lenz's Law

Lenz's law provides the direction of the induced current that results from a changing magnetic flux. It states that the induced current will flow in such a direction that the magnetic field created by this current opposes the change in the original magnetic flux. This law ensures that the induced effects act to counter the variation that produced them.

Transcript

00:01 We have a circular loop of area a it is placed perpendicular to the time wearing magnetic field b t equals to b0 plus a t plus b t square where b not a and b these are constant values okay so for the part a of the question we have to determine the magnetic flux at t equals to 0 okay so the magnetic flux 5 is given by the magnetic flux 5 is given by b a cost theta and theta is the angle between magnetic field and area vector so since coil is placed the loop is placed perpendicular to the magnetic field so this theta will be 0 so from here we will get b multiplied by a cos 0 degree and cos 0 is 1 so we will get b multiplied by a so at t equals to 0 b at t equals to 0 from this expression we will get these value equals to 0 so we will get b not only so phi at t equals to 0 it will be equals to b which is equals to b not so b not multiplied by area a so this is the answer for the part a of the problem okay now moving to the next part b in which we have to determine the equation for the induced potential difference delta we induced in the loop as a function of time so it is given by minus d by d t so from here we will get minus d by d by d t of b multiplied by area and b is changing with time so area is constant so we can get that delta v induced it will be equals to minus a and d by d t of b which is b not plus a t plus b t plus b t square okay so differentiating this term with respect to time and after solving we will get delta b induced it is equals to minus a and bracket a plus 2 b t okay so this is the answer for the induced potential difference in the loop okay with respect to function of time t now moving to the part c in which we have to determine the magnitude and direction of the induced current so induced current i it will be given by delta v induced divided by resistance r so we are looking for the magnitude so we can take the modulus okay so uh delta v induced it is equals to minus a a small a plus 2 b t divided by resistance r.

02:24 So from here we will get that induced current i, it is equals to area a small a plus 2 b t divided by resistance r and this current will be in the counterclockwise direction counterclockwise direction...

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