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Problem 22 Medium Difficulty

An $N$ -turn circular wire coil of radius $r$ lies in the xy-plane (the plane of the page), as in Figure $\mathrm{P} 20.10$ . A uniform magnetic field is turned on, increasing steadily from 0 to $B_{0}$ in the positive $z$ -direction in $t$ seconds. (a) Find a symbolic expression for the emf, $\boldsymbol{\varepsilon},$ induced in the coil in terms of the variables given. (b) Looking down on at the $x y$ -plane from the positive $z$ -axis, is the direction of the induced cur- rent clockwise or counterclockwise? (c) If each loop has resistance $R$ , find an expression for the magnitude of the induced current, $I .$

Answer

a. \frac{\pi r^{2} N B_{0}}{\Delta t}
b. \text { clockwise }
c. \frac{\pi r^{2} B_{0}}{R \Delta t}

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Video Transcript

for part A were asked to find a symbolic expression for the induced IMF and the coil in terms of the variables that are given so well indicate this is part of a So the induced e m f is equal to the number of turns in the coil multiplied by the magnitude of the changing magnetic flux. I felt if I divided by the change in time, felt a t so this is equal to the number of turns multiplied by. Well, the area of the coil is not gonna change right. The only thing that's changing is the magnetic field from the initial to the final. So this is the final magnetic field multiplied by the area, multiplied by the co sign of the angle. Fada minus be initial multiplied by the area multiplied by the same angle coastline data all divided by Delta teak. So all that's over felt a T Well, we're told here that the final magnetic field is equal to be not. We're told that initially the magnetic field is equal to zero. Based upon the setup of the problem, Veda is equal to zero degrees. We're told that the time that transpires is just time t so Delta t here it's just equal to t. So therefore, the induced e m f is going to be equal to the number of turns. Times be zero, right? Because the initial zeros that expression goes away, That part goes away. The area is pi r squared since it's circular and then co sign of zero is one. So we can just ignore that term. And this is all divided by t. So this is our expression for the induced in meth and we can go ahead and box it in Is their solution for part A for part B, it says looking down on the X Y plane from the positive Z access is the direction of the induced current clockwise or is it counterclockwise? Okay, so the direction of the magnetic field is upwards, which is the positive ZY direction. The coil is viewed from the positive Z access. So in this case, the current induced in the coil must appear to be the in a clockwise direction. This is because the magnetic field is increasing in the positive direction and due to which the link magnetic flux also directed in the positive dpz direction. So to oppose this flux, the induced current should produce the magnetic field in a downward direction or the negative Z direction. Therefore, the induced current must flow in a clockwise direction. So for part B, we can say that the magnetic field is in the Z hat direction. Therefore, the, um therefore, the induced current must flow in a clockwise direction. So thus we can say thus induced current in clockwise direction. We can go in and box that in as our solution for part B. Okay, Now, for part C were told that if each loop has a resistance, are finding expression for the magnitude of the induced current. Okay, well, the induced current I is equal to the induced at the M f divided by the resistance. We're gonna call this our prime, and we're told that our prime is equal to the number of terms times the resistance in each turn. Okay, So, plugging that value into this expression, we find that I concert this down just a little bit more so we have some more room. We find that I is equal to what we, uh the induced e m f which we found on the first page to be the number of turns times B zero times pi r squared, divided by time But of course, the number of turns are gonna cancel out because our prime also has the same number of turns in it. So we're left with an expression that's be zero times pi r squared, Okay. Divided by the time that transpires t multiplied by the resistance in each turn are we can go ahead and box set in as our expression for the current in the answer for part C.

University of Kansas
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