00:01
Okay, this is question 2862, and we have two long wires that lie along this y -axis that are pointing into the page on the bottom one and out of the, into the page in the bottom and out of the page in the top wire.
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And we want to figure out the magnetic field at point p over here.
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So first, part a asked us to draw the.
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Vectors to show the direction of the magnetic field coming from each wire and then find the net magnetic field.
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So let's think about this first wire up here.
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To find the direction of the magnetic field, we want to use the right -hand rule.
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So to do that, let's point our thumb in the direction of the current, which is out of the page, so pointing out of the page to us.
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And think about what direction our fingers are curling.
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At this point p right here.
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And if we think about that, we're gonna get some field pointing out like this.
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So this is a contribution from the top wire here.
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Let's do the same thing for the bottom wire, where we point our thumb into the page and curl our fingers around.
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That's gonna give us something pointing down into the right like this.
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So this is from the top wire.
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This is from the bottom wire.
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I'll call two.
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So now we just need to add them together to find the net magnetic field.
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And when we add these two together, since the currents here are equal, the magnitudes, the length of the vectors are going to be the same.
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And so the y components are going to cancel out, which means we're only going to add the horizontal components.
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So that's going to give us a big net vector pointing straight to the right like this.
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So this is going to be the direction of the net magnetic field.
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Now part b asks us to find an expression for the magnitude of the magnetic fields at any point on the x -axis in terms of the x -cordinate of the point.
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So what we're going to do here, we need to use our expression.
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B is going to be from one wire.
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We would have, going to have a factor of mu not times pi divided by 2 pi r.
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So this is going to be the expression for r.
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And then we're going to need a factor of sine theta to take into account this angle.
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So this is going to be the expression for the magnetic field.
02:59
Coming from one of these wires at a point on the x -axis because the x -cordinate here is going to be represented or sorry the the distance the y -cordinate here will be represented by this sine theta now we have two two wires here so what we're going to want to do is just multiply this by a factor of two to get the total net magnetic field and this angle because of the symmetry here these are both still separated by a distance a vertical distance a.
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This angle theta on the bottom is going to be the same.
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So we can just simply multiply this by two.
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That's going to give us, we're going to make these factors of two will cancel out.
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So we're just going to be left with this as our expression.
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Now we don't actually know what sign of theta is.
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So that's going to be our next step.
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If we think about this triangle here, the sign of any angle is the opposite angle over the hypotenuse.
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So we need to find the hypotenuse of this triangle.
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And to do that, we just want to use the pythagorean theorem.
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So using the pythagorean theorem gives us the squared of a squared plus x squared.
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So that's going to be the length of this total distance along the hypotenuse of this triangle here.
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So now remember sign is going to be the opposite, which is a over the hypotenuse.
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So i'm just going to plug that in right here.
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So opposite is a and then divided by the hypotenuse, which is this whole thing.
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So this is.....