The rectangular loop in Figure 21.66, with area A and...

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.

Key Concepts

Time-Dependent Sinusoidal Behavior

When a loop rotates in a uniform magnetic field, the angle between the field and the normal to the loop changes sinusoidally. This causes the magnetic flux to vary as a cosine function of time, which in turn leads to its derivative, the rate of change of flux, demonstrating a sinusoidal behavior shifted by a phase. This relationship is key to predicting the time-varying induced emf in such systems.

Effect of Angular Velocity on Induced EMF

The angular velocity of a rotating loop directly influences the rate at which the magnetic flux changes. An increase in angular velocity results in a faster rate of change of flux, thereby increasing the magnitude of the induced emf, as indicated by Faraday's Law. This relationship highlights how the dynamics of rotational motion can be used to control and modulate electromagnetic induction.

Magnetic Flux

Magnetic flux is a measure of the total magnetic field passing through a given area. It is defined as the dot product of the magnetic field vector and the area vector. In contexts where the area’s orientation changes with time, such as a rotating loop, the flux becomes a time-dependent quantity that often involves a trigonometric (cosine) function reflecting the change in angle between the magnetic field and the area vector.

Faraday's Law of Electromagnetic Induction

Faraday's Law states that an electromotive force (emf) is induced in a closed circuit when the magnetic flux through the circuit changes over time. The magnitude of the induced emf is equal to the negative rate of change of magnetic flux with respect to time. This principle is fundamental in understanding how varying magnetic environments can create electrical currents in conductors.

Transcript

00:02 In this problem, we have a rectangular region.

00:05 I'm drawing that with orange, and that is rotating at a uniform angular velocity omega.

00:11 It's rotating in a uniform magnetic b field that's pointing along the x -axis, and at time t equals zero, this region is entirely contained in the yz plane.

00:25 Our goal is to find the plots of the flux, the rate of change of the flux, the emf, and the emf, if we were to double the angular velocity.

00:35 And i'm going to draw all of those on the same plot.

00:38 So let's start out with a flux.

00:40 Let's recall that the net flux in a uniform b field is given by ea cosine, where fee is the angle between the perpendicular of the area and the b field.

00:55 So the perpendicular of our area, i'll draw that in red on our diagram, and that's given by that guy.

01:02 So we see at time t equals zero, these are pointing in the same direction.

01:08 So we don't have to worry about any phase shift.

01:10 This is just b a cosine omega -t.

01:15 On our plot, we see that this has an amplitude of b -a.

01:20 So i will draw this in green.

01:25 And on the axes, i'll delineate that by, let's go pi over two omega, pi over omega, 3 pi over 2 omega, and we'll go 2 pi.

01:42 So we just have a normal cosine here.

01:45 Let's see how we can do this.

01:49 Zero, maximum negative, zero.

01:55 Okay, for the second part, we have the rate of change of our flux.

02:05 Now, this is going to be given by the slope of the plot that we just drew.