A long straight wire lies along the y axis and carries a...

A long, straight wire lies along the $y-$axis and carries a current $I =$ 8.00 A in the $-y$-direction ($\textbf{Fig. E28.21}$). In addition to the magnetic field due to the current in the wire, a uniform magnetic field $\overrightarrow{B}_0$ with magnitude 1.50 $\times$ 10$^{-6}$ T is in the $+x$ direction. What is the total field (magnitude and direction) at the following points in the xz-plane: (a) $x =$ 0, $z =$ 1.00 m; (b) $x =$ 1.00 m, $z =$ 0; (c) $x =$ 0, $z = -$0.25 m?

Transcript

00:01 Here we're going to be finding the total magnetic field due to two sources, one of which is a wire, so we'll need ampher's law, which you may use to find that the magnetic field due to a wire in its magnitude is a product of the fundamental constant permeability of free space, mu nod, times the current in the wire over 2 pi.

00:31 Times the distance from the wire.

00:35 The direction of that is a circle going around the direction of the wire, figured out by using your right hand, and we'll do that in a little bit.

00:49 But we need some points to find the total magnetic field, and we will have to use superposition.

00:57 That is the magnetic field total, total is a sum vector of the uniform b0 that points to the x plus the magnetic field of the wire.

01:15 And the direction of that magnetic field of the wire will depend on what position you're looking at.

01:22 So there are several positions that have been identified.

01:26 One is one meter away from the wire, which we put right down the y -axis going through.

01:33 The x -z plane at the origin.

01:38 The other one is one meter out along the x direction.

01:45 And finally, there is a point c in the negative c direction, 0 .25 meters behind what we're looking at in the negative c direction.

02:07 Okay, so we have two magnets.

02:09 To figure out for the b wire, but let's figure out the direction.

02:13 So the right -hand rule, usually what i do is i put my thumb in the direction of the current of my right hand, and then i see which way my fingers curl around the wire.

02:31 And so what i see is in the front on the z -axis, the wire is going back to the negative.

02:40 X.

02:42 On the other side, it's going to the positive x.

02:47 And it comes out in the z direction, in the front, and goes into the minus c back behind.

03:03 So that's going to be important when we think about those three points in the magnetic field of the wire.

03:09 So let's get to it.

03:12 For point a, we're going to figure out the magnitude of the b wire is mu -0, which is 4 pi times 10 to the minus 7, 8 constant, times the current, which, let's see, they give that as 8 amps.

03:42 I think i wrote that down.

03:49 Let's do that 8 amps.

03:55 So it's a substantial current...

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