3. Consider a circular coil with 30 turns of wire and a...

3. Consider a circular coil with 30 turns of wire and a radius 4.0 cm. A magnetic field B is perpendicular to the plane of the loop and its magnitude is changing with time according to B(t)=B_0 cos(wt) ,where B_0=0.35 T and w=24.0 s^-1 . a) What would be the maximum induced voltage in the coil? b) What is the first positive time when this maximum occurs? c) Sketch the voltage across the coil as a function of time. d) The coil has a resistance of 1.5 ? . Use Ohm's law to calculate the maximum value of the induced current.

Transcript

00:01 So we're looking at a magnetic field going through a coil of wire, which means we're going to have to deal with magnetic flux.

00:10 So we have 300 turns of wire, and we have our magnetic field going through here.

00:19 And that magnetic field is given by this time -dependent equation.

00:24 E0, cosine omega -t, or b -0, whatever you'd like to call it.

00:30 We have some values to work with here.

00:32 This is, i believe, 350 militaire.

00:38 4 .35 tesla.

00:43 And our omega is 24 radiance per second.

00:52 And we know that the radius these coils make is 4 centimeters.

00:59 We want the maximum voltage going through here.

01:03 So we need to look at our induced emf, which is going to be dependent on the magnetic fluxes time derivative.

01:13 And when we're looking at magnetic flux, when the magnetic field is perfectly perpendicular to the plane of the coil, it's just magnetic field times the area created by that coil.

01:26 And that area is going to be area of a circle, pi r squared with this given radius.

01:32 So we have our 350 million tesla zero.

01:37 We have cosine.

01:40 And our area is pi, and we're going to put 4 times 10 to the minus 2 square that because we're going to keep it in meters.

01:50 Okay.

01:52 Let's go ahead and take the time derivative because there's only one time -dependent portion of this, right? and that says cosine 24 -t.

01:59 We take the time derivative of that.

02:01 We take the derivative of respect to t of the term inside cosine.

02:05 So that's just 24.

02:07 And then the derivative of cosine, which is minus sign.

02:11 So if we have a negative time derivative of this from the minus n in front and we end up with a negative 24 in front, those negatives are going to cancel.

02:22 And our emf is looking like 300 because that's the number of turns, 24 from that derivative, and then the rest of it.

02:32 We have a sine 2 .4 t.

02:36 We have pi and 4 .10 to the minus 2.

02:43 It's 4 centimeters.

02:46 Put that all together...

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