A closely wound, circular coil with radius 2.40 cm has 800 turns. (a) What must the current in

A closely wound, circular coil with radius 2.40 cm has 800...

Key Concepts

Magnetic Field of a Circular Coil

This concept involves calculating the magnetic field produced at the center of a circular coil. It uses the well-known expression B = (??NI)/(2R), where N is the number of turns, I is the current through the coil, R is the radius, and ?? is the permeability of free space. This formula is derived using the Biot–Savart Law and demonstrates how the geometry and current in the coil contribute to the strength of the magnetic field at its center.

Biot–Savart Law

The Biot–Savart Law is a fundamental principle used to calculate the magnetic field generated by a current element. It provides the basis for deriving field expressions for various current configurations, including circular loops. The law takes into account the distance from the current element to the point of interest and the sine of the angle between the current element and the line joining the element to the point, making it essential for deriving the field contributions of each segment of the coil.

Magnetic Field Along the Axis of a Circular Coil

This concept addresses how the magnetic field varies with distance along the axis perpendicular to the plane of the coil. The field at a point on the axis is given by B(x) = (??NI R²)/(2(R² + x²)^(3/2)). This formula illustrates that the field strength decreases as one moves away from the center of the coil and is crucial for solving problems where the field is required to be a specific fraction, such as half, of its maximum value at the center.

Transcript

00:01 Okay, we're looking at question 28 .36, and we have a circular coil with radius 2 .4 centimeters, and we're looking for the current in the wire of the magnetic field at the center of the coil is 0 .077 tesla.

00:18 So we're trying to solve here for the current.

00:23 So to use this, we're going to need to use this formula for the magnetic field along the axis of, a circular loop of wire.

00:34 So we have on the top, u knot times n, the number of turns, times the current, and a is the radius here.

00:42 And i've listed the quantities that we have over here and converted them to si units.

00:47 I've converted the 2 .4 centimeters to meters to meters.

00:50 So we have that readily available to us.

00:53 And then that's divided by two times x, where x is the distance from the center along the axis of the coil.

01:03 And then a again is the radius, and that's raised to the power of three halves.

01:07 So when we're first looking in part a here, we're looking at the center of the coil.

01:12 So at the center of the coil, x is equal to zero.

01:17 So we can get rid of that term when we actually start plugging things in here.

01:25 So when we look at this, we're going to end up with a factor of mu not up at the top when we start plugging this n times n times i and then we have our a squared now let's look at the bottom here with x equal to zero when x is equal to zero we have two times then this factor of a squared raised to the power of three halves and what that does a squared to the three halves is going to simplify to just a cubed if we take a look at this we have a squared on top and a cubed on the bottom and that's going to cancel out to just a simple factor of a in the denominator so we can write this easily as this and this i'm going to just label this b c because this indicates the magnetic field at the center of the at the center of this coil so technically what we're given here is b c the magnetic field at the center so now all we have to do is rearrange this to solve for i because i is what we want.

02:36 So we're going to end up with 2 times a times bc here when we multiply both sides by bc.

02:43 That's going to be equal to our mu not times n times i.

02:48 And then to solve for i, we're going to just divide both sides by this factor mu not times n.

02:55 So we get over here, mu not times n, and this is our i.

03:01 I'll just write this on the other side.

03:04 So this is what we want to solve for.

03:06 Now we just have to plug in.

03:07 A is 0 .024, b .c.

03:11 0 .077, our factor of mu not, and n is 800.

03:15 So when we plug that all in, we get i is equal to 3 .68 ampires.

03:22 So that's the answer to part a.

03:25 Now, for part b, we're asked at what distance x from the center of the coil on the axis of the coil.

03:32 So this is our x here that's coming in.

03:35 When is the magnetic field half its value at the center? so, all right, now i'm going to pull this over here.

03:43 This we figured out from part a.

03:47 So we want to figure out, basically we're looking for some distance.

03:52 Some magnetic field, f x, where it's fc, the magnetic field at the center, divided by two.

04:01 This is what we're trying to solve for.

04:03 And our bx is going to be this big expression here.

04:07 So this is really our generalized bx, because this is at some distance x along the axis.

04:14 So we have this big equation up here.

04:18 Looks pretty complicated.

04:20 But then take a look at b .c.

04:24 We have this slightly simpler version...

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