Two very, very long, straight current-carrying wires carry...

Two very, very long, straight current-carrying wires carry equal currents I in opposite directions, parallel to the y-axis. One wire lies along the y-axis, and the other lies 2.00 cm away to the left, along the line x = ? 0.0200 m. (a.) If the magnetic field due to these two currents at the point (x,y) = (0.120 m, 0) is B?? = 4.40 × 10?? (+?) T, what is the magnitude and direction of the current that lies along the y-axis? I = A, in the direction (b.) Consider the same two parallel wires, in the same positions, carrying equal currents I in opposite directions parallel to the y-axis. But for part (b.), please assume the current through each wire has magnitude I = 6.00 A. What is the magnitude and direction of the magnetic force per unit length on the current-carrying wire that lies along the y-axis, in micro-Newtons per meter (?N/m)? FB/? = ?N/m, in the direction

Transcript

00:01 All right, hello.

00:01 In this question, we're given this setup where we have two long current -carrying wires and we're told they're carrying currents in opposite directions.

00:10 One of them, the blue wire, is located on the y -axis, so that's going to be at x equals zero.

00:14 The other wire is located to the left, a distance of two centimeters.

00:18 And then we're told at a point 12 centimeters to the right of the y -axis, we have a magnetic field coming out of the page in the positive k direction, and we're told what that magnitude is.

00:32 With that information, we're asked to figure out what is the direction in magnitude of the current -carrying wire on the blue wire on the axis there.

00:41 We know that the magnitude of the current is going to be the same in both, which is going to be very useful, but we need to figure out what the direction is going to be.

00:49 And to do that, i'm going to use the fact that our magnetic field created by a wire is equal to mu -naught times i of that wire over two pi times r, where r is our distance away from our point of interest.

01:03 So i know that my magnetic field here, b, is coming out of the page on the right -hand side.

01:11 And since i know that's closer to the blue wire than it is to the red wire, i know that the blue wire is going to have a greater influence at this point.

01:19 So i can say that the b of, i'll call it the blue wire, i know that that is going to be greater than the magnetic field created by the red wire, which i guess is an r, not a b.

01:31 And so in order to figure out the direction of the currents, well, i'm going to say i want my net magnetic field to be out of the page.

01:37 So if i take my right hand and i put my thumb down and run it along the blue wire and curl my fingers, i can see that the blue wire is going to be creating a magnetic field that comes out of the page on its right side.

01:51 And if i do the same thing for my red wire, i'll find that it generates a magnetic field going into the page.

01:57 But that end of the page field is going to be weaker.

01:59 So because my magnetic field net is coming out, i know that the direction of the current in the blue wire must be such that this setup is true, and that's only true if the current is going down.

02:11 So that's why i've drawn the currents as i have.

02:13 So we know that the current in the blue wire is going to be going down.

02:16 Now i just need to figure out what the current is going to be.

02:18 Well, i know that this here, my net magnetic field here, is going to just be my magnetic field generated by my blue wire.

02:27 And then in the opposite direction is going to be my magnetic field generated by the red wire.

02:31 And i know that these are going to both be mu naught i, and i is going to be the same, over 2 pi, but my radii are going to be different here, right? so radius of the blue minus mu naught i over 2 pi radius of the red.

02:45 I can go ahead and simplify that down a little bit, and i'm going to have mu naught i over 2 pi times 1 over the radius of the blue wire minus 1 over the radius of the red wire.

02:58 The radius of the blue wire in this case is going to be this distance, 0 .12 meters, and the radius of the red wire is going to be this distance, which is 1 .2 plus 0 .02.

03:11 So i have all of these values, so i can go ahead and plug them in, though i just realized i don't want to find b, i want to find i, right? so i'm going to go ahead and solve this equation for i...

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