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The rectangular loop in Figure $21.66,$ with area $A$ and resistance $R,$ rotates at uniform angular velocity $\omega$ about the $y$ axis. The loop lies in a uniform magnetic field $\vec{B}$ in the direction of the $x$ axis. Sketch graphs of the following quantities, as functions of time, letting $t=0$ when the loop is in the position shown in the figure: (a) the magnetic flux through the loop, (b) the rate of change of flux with respect to time, (c) the induced emf in the loop, (d) the induced emf if the angular velocity is doubled.

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a. Graph of the magnetic flux through the loop is as follows,graph not availableb. Graph of the rate of change of flux with respect to the time is as follows,graph not availablec. Graph of the induced emf in the loop is as follows,graph not availabled. Graph of the induced emf if angular velocity is doubled is as follows,graph not available

Physics 102 Electricity and Magnetism

Chapter 21

Electromagnetic Induction

Current, Resistance, and Electromotive Force

Direct-Current Circuits

Magnetic Field and Magnetic Forces

Sources of Magnetic field

Inductance

Alternating Current

Cornell University

Rutgers, The State University of New Jersey

McMaster University

Lectures

03:27

Electromagnetic induction is the production of an electromotive force (emf) across a conductor due to its dynamic interaction with a magnetic field. Michael Faraday is generally credited with the discovery of electromagnetic induction in 1831.

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In physics, a magnetic field is a vector field that describes the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude (or strength); as such it is a vector field. The term is used for two distinct but closely related fields denoted by the symbols B and H, where H is measured in units of amperes per meter (usually in the cgs system of units) and B is measured in teslas (SI units).

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and this problem. We have a rectangular region. I'm drawing that with orange and that is rotating at a uniform angular velocity omega. It's irritating in a uniform magnetic Byfield that's pointing along the X axis and at time T equals zero. This region is entirely contained in the Y Z plane. Our goal is to find the plots of the flux, the rate of change of the plucks, the IMF and the N f if we were to double the angular velocity, and I'm gonna draw all those on the same plot. So let's start out with the flux that recalled that the net flux in a uniformed Byfield. It's given by E a co sign fee where he is the angle between the perpendicular of the area and the Byfield. So the perpendicular of our area. I'll draw that in red on our diagram, and that's given by that guy. So we see at Time T equals zero. These were pointing in the same direction, so we don't have to worry about any phase shift. This is just be a co sign Omega T on our plot. We see that this has an amplitude of B A. So I will draw this in green and on axes. I'll delineate that by pi over to omega High over omega three pi over to Omega. I don't go to so we just have ah normal co sign here. She could do this. Zero, uh, actually negatives You. Uh okay, for the second part, we have the rate of change of our flags. Now, this is gonna be given by the slope of the plot that we just drew. So let's let's get a qualitative feel for what they should be. We see that it starts out with zero slope, so it's gonna have a note here as a zero sum pi over a mega. So it is zero here as a zero slope at the end, so we know that it must be zero at those points. We also see that it's maximally negative right there and maximally positive right there so they actually get the amplitude of this thing. We would need calculus, and that tells us that this is equal thio minus omega be a sine omega t. So, assuming that omega is greater than one, we'll have this imp itude a little bit larger and not just you being a B a And now we can see how our qualitative field got us most of the way. There we start out at zero. We go to maximally negative, go back to zero, go to maximally positive and we go back to zero for the next part. We want Thio plot the and for the enough When we're plotting, we want to recall the sign Convention. The IMF is minus the rate of change of our flux. So this is just gonna be what we just drew but inverted so mega be a sine omega t This one all draw on blue So goes from zero now maximally positive zero maximally negative zero for the last one We're being asked what if Omega is doubled? So and see, we just replace these amigos with two omega and we have the IMF is to omega e a sign to Omega T. So now we see the amplitude is gonna double to omega B A and also the angular frequency doubling means that it's gonna have a shorter period. So let's delineate some middle marks here now putting it into the boards. So the angular frequency doubling means that the period is cut in half. So now for the plot that we just drew in blue, where this has a negative and I'll draw this last one orange where this has a maximum, that maximum is going to occur at the halfway point and likewise for every single point on that plot. So let's see what this looks like. It goes from zero to maximum 20 I'm actually negative to zero, actually positive to zero. Next thing you knew to zero. All right, those air in those of the four plots.

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