In each of six electric motors, a cylindrical coil with N...

In each of six electric motors, a cylindrical coil with $N$ turns and radius $r$ is immersed in a magnetic field of magnitude $B$. The current in the coil is $I .$ Rank the motors in order of the maximum torque on the coil, greatest to smallest. (a) $N=100, r=2 \mathrm{cm}, B=0.4 \mathrm{T}, I=0.5 \mathrm{A}$ (b) $N=100, r=4 \mathrm{cm}, B=0.2 \mathrm{T}, I=0.5 \mathrm{A}$ (c) $N=75, r=2 \mathrm{cm}, B=0.4 \mathrm{T}, I=0.5 \mathrm{A}$ (d) $N=50, r=2 \mathrm{cm}, B=0.8 \mathrm{T}, I=0.5 \mathrm{A}$ (e) $N=100, r=3 \mathrm{cm}, B=0.4 \mathrm{T}, I=0.5 \mathrm{A}$ (f) $N=50, r=2 \mathrm{cm}, B=0.8 \mathrm{T}, I=1 \mathrm{A}$

In each of six electric motors, a cylindrical coil with $N$ turns and radius $r$ is immersed in a magnetic field of magnitude $B$. The current in the coil is $I .$ Rank the motors in order of the maximum torque on the coil, greatest to smallest.
(a) $N=100, r=2 \mathrm{cm}, B=0.4 \mathrm{T}, I=0.5 \mathrm{A}$
(b) $N=100, r=4 \mathrm{cm}, B=0.2 \mathrm{T}, I=0.5 \mathrm{A}$
(c) $N=75, r=2 \mathrm{cm}, B=0.4 \mathrm{T}, I=0.5 \mathrm{A}$
(d) $N=50, r=2 \mathrm{cm}, B=0.8 \mathrm{T}, I=0.5 \mathrm{A}$
(e) $N=100, r=3 \mathrm{cm}, B=0.4 \mathrm{T}, I=0.5 \mathrm{A}$
(f) $N=50, r=2 \mathrm{cm}, B=0.8 \mathrm{T}, I=1 \mathrm{A}$

Key Concepts

Loop Area

The area of the loop is a critical parameter in determining the torque and is calculated as A = ?r², where r is the radius of the cylindrical coil. Since the torque is directly proportional to the area, even small changes in the radius, due to the square relationship, can significantly affect the torque produced.

Magnetic Torque on a Current Loop

This concept involves understanding that a current-carrying loop placed in a magnetic field experiences a torque. The maximum torque occurs when the plane of the loop is perpendicular to the magnetic field, and is given by the formula ?_max = N I A B, where N is the number of turns, I is the current, A is the area of the loop, and B is the magnetic field strength.

Proportional Relationships in Coil Parameters

The maximum torque on the coil is directly proportional to the product of the number of turns (N), the current (I), the magnetic field strength (B), and the area (A, which depends on the square of the radius). Understanding this linear relationship helps in comparing different coils or motors by evaluating the cumulative effect of these parameters on the torque.

Transcript

00:01 Hi everyone.

00:03 The concept we're going to look at in this video is torque on a current loop.

00:08 All right.

00:09 And so the torque on a current loop is tau.

00:14 And tau is equal to the number of turns in the loop times the current in the loop, times the area of the loop, times the magnetic field the loop is sitting in.

00:27 Okay.

00:28 And so we can say the area, since it's a loop, is pi.

00:34 Times r squared.

00:38 All right.

00:38 Now we've got six lutes.

00:42 And let's just make a table.

00:45 So we've got two of them with 100 turns.

00:54 The next one has 75 turns.

00:59 Next one is 50 turns.

01:04 Another one that's 100 turns.

01:07 And then 50 turns.

01:09 Okay, the current is in amps, and these are all at 0 .5, except for the last one, which is at 1.

01:21 0 .5, the last one is 1.

01:35 Okay, let's put the magnetic field next, and the magnetic field is in tesla's capital t, and these kind of vary.

01:48 So the first one is 0 .4 teslas.

01:52 The next one is 0 .2 teslas, then back to 0 .4 teslas, then 0 .8 teslas, 0 .4 tesla's, and the last one's 0 .8 tesla's.

02:10 All right.

02:12 And then we have the r's, and we might as well make a pi r squared or let's just list the r's.

02:21 Okay, so these are all in centimeters, so they'll all need to be divided by 100.

02:29 So the first one is two centimeters, next one is four centimeters, two centimeters, two centimeters, three centimeters in radius, and finally two centimeters in radius.

02:48 All right, so let's multiply n i and b.

02:55 So when we do 100 times 0 .5 times 0 .4, we get 20.

03:04 Okay.

03:05 100 times 0 .5 times 0 .2 is 10...

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