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As shown in the figure below, two Iong parallel wires (1 and 2) carry currents of I1 3.10 and I2 5,20 in the direction indicated_(a) Determine the magnitude and dir...

Question

As shown in the figure below, two Iong parallel wires (1 and 2) carry currents of I1 3.10 and I2 5,20 in the direction indicated_(a) Determine the magnitude and direction of the magnetic field at point midway between the wires (d 10,5 cm) magnitude direction counterclockwise from the +x-axis(b) Determine the magnitude and direction of the magnetic fleld at point P located d 10.0 cm above wire magnltude dlrectlon counterclockwlse from the +X-axls

As shown in the figure below, two Iong parallel wires (1 and 2) carry currents of I1 3.10 and I2 5,20 in the direction indicated_ (a) Determine the magnitude and direction of the magnetic field at point midway between the wires (d 10,5 cm) magnitude direction counterclockwise from the +x-axis (b) Determine the magnitude and direction of the magnetic fleld at point P located d 10.0 cm above wire magnltude dlrectlon counterclockwlse from the +X-axls



Answers

Two long, straight, parallel wires are $10.0 \mathrm{~cm}$ apart and carry 4.00 A currents in the same direction (Figure 20.65 ). Find the magnitude and direction of the magnetic field at (a) point $P_{1}$, midway between the wires, (b) point $P_{2}$ $25.0 \mathrm{~cm}$ to the right of $P_{1}$

Here we have two long parallel wired give that are carrying given currents. And we also know the directions cousins there indicated for us in the figure. Now we want to find the magnitude and direction of the magnetic field at a point midway between the wires first. So in this case, we start by realizing that since the since the currents are going in opposite directions being that must be equal to be one minus B two. So we find that you know it must be must be equal that are sort of refined, that being that must be equal to muna times, I minus I to over two pi r from the equation for the magnetic field from a infinite long wire. Now we can move on and when we plug in our values, we find that Dina is equal to negative four time for micro Tesla which we couldn't find using the right hand rule must be directed towards the bottom of the page. Now, the second part of the problem asked us to find the magnitude and direction of the magnetic field in a point P, which is a certain distance above the weir on carrying a given current so we can draw out arc waken, draw out our wires and are currents as well. The magnetic field like this, we can see that are one must be equal to 0.2 times the square root of two by trigonometry, and it's directed at um state. A one region equal to 135 degrees are too must be equal to 0.2, which is directed at 180 degrees. Now all that's left for us to do is to plug it. So once again, we're going to use the equation for the magnetic field from an infinitely long wire. And when we plug in our given current as well as the radius, that would give it in the problem, we find that V one is equal to 2.12 Micro Micro Tesla and B two is five micro Kessler. Now we could find the magnetic fields in the X and Y directions or that other components. So being an X is equal to be one times coast of 1 35 plus B two times coast of 1 80 which is going to give us a negative 6.5 my car. Tesla being that why is gonna be equal to be one times sign of 1 35 plus zero since the sign of 1 80 just gives us zero way thought that and we find that is equal to 1.5 micro Tesla. Now our complete be net, including all components, must be equal to our X component squared plus R Y component squared. We plug in our results from earlier. We find that this is equal to 6.67 Micro Tesla to find the angle that is directed at, we can use the fact that this must be equal to tan to power of negative one or arc tanne off the X over V Y, which is gonna give us 167 degrees. So our final answer is a field of 6.67 micro Tesler times directed in the direction of 100

Questionnaire asked to consider too long parallel wires. The first wire has current I one of three amperes. The second wire has current I two of five appears so the magnetic field from a long straight wire I right here, it's B is equal to mu, not times the current divided by two pi r r r is the distance from the wire that carries current I to the point we're trying to calculate the magnetic field. That and d here, we're told, is midway between the two wires is the distance D, which is 20 centimeters or 0.2 meters for part. It wants us to calculate the magnetic field at that point between the two wires so indicate this is a I want to do this. We're getting the net. Magnetic field will call it beasts of End for being met. And it's equal to the magnetic field of the second wire since its larger minus the magnetic field of the first wire. Okay, since they're going in the same directions, their opposing right, so we'll just plug in our values. Here we have you, not times I, too. For the second wire, divided by two write that to make it look nicer two times pi times are but are here is the distance that's halfway between the two wire. So it's d It's d divided by two. We're gonna subtract from that B one B one is mu not times. Uh, I won divided by two times pi attempt the distance for B one would be also be over too. Okay, so, um, you not here is a magnetic permeability of free space is equal to four pi times 10 to the minus seven. Plugging that as well as our other values. In this expression, we find that this is equal to four times 10 to the minus six Tesla. Okay, So to find the directionality, we can use the right hand rule. But the question that what we need know is which magnetic field is gonna overtake, right. So to use the right hand roll, you point your thumb in the direction of the current. You point your fingers of your right hand and direction of the magnetic field that you're tryingto you put your fingers in the direction of the point you're trying to calculate. So the point where the point p at the distance D over two away. You're gonna put your fingers in that direction while your thumb points in the direction of the current, and then you curl your fingers, whatever direction your fingers curl. That's the direction of the flow of the magnetic field, where you can see that B two and B one are in the opposite directions here. But since B two is greater than be one, we can use the right hand rule for B two to determine the direction of the flow so long, just so he too, is greater than be one. The reason that's true is because I, too, is greater than I want. And that's the only difference in B two and B one is the current. And because of that we can determine using the right hand rule from Ah, wire, too. And we use that to determine that this is downward. Then we can box. All this in is our solution. For part A be says, find the magnitude and direction of the magnetic field at a point P located a distance D 20 centimeters above the wire, carrying the current of five amperes. Okay, so we'll start a new page for that to get be here. So I drew a little diagram in paint. Uh, that I think nicely shows what they're trying to show here. So let me pull this diagram, this little file out of paint here and I'll import it. So that way you can see what I'm talking about. Okay, so this is the situation that they're describing here. All right? Where it's that Point p. And so we have this angle 45 degrees that's being made with wire I won and that England's of being the same angle. 45 degrees for vector. Be one from the magnetic field. Remember, magnetic fields are vectors because they have direction. All right, so now again, we need to calculate in that magnetic field. But first, let's calculate the distance for our want Our one is gonna be equal to, According to this figure, the square root using Pythagorean theorem, right? Are one here, look, this is this distance. Here are one indicate that. So this is just gonna be d squared plus D squared Cordy a 0.2 meters. So carry out this operation. We find that our one is equal to 0.28 meters. Where are two is just obviously D so D Here is also this is also our two. All right, so now, to get the net magnetic field, we need to first calculate the magnitude of the magnetic field for B one and B two. Since the angles are different, things are gonna be a little different than what we did in part ex impart a the angles for B one and B two or the exact same. So we don't have to worry about this. So be one is mu not I would over to pie are one this comes out to equal says I won. So this comes out sequel to 0.14 times 10 to the minus six Tesler and be too magnitude again. This is magnitude is mu not times I too divided by two times pi and then the distance are too. There is the same as the distance deep plugging in that value we find this magnitude is equal to five times 10 to the minus six Tesler. Okay, so now to get the net magnetic field, we're gonna calculate the vector and then we can use that use the X and y components and the dagger and the room to get the magnitude of the vectors. So the vector be net is equal to vector B one plus sector B two. Okay, so we're gonna put B one and B two in there X and Y components. So the X component of B one is B one times. And if you look back to our figure here, it's gonna be be one times fee sign of 45 degrees. So you have to be careful because sign is opposite over hi partners. So this is gonna be one timesthe sign up 45 degrees One of the times we just automatically associate co sign with the ex direction. But it all depends upon where you're defining your angles. And this is in the defined as the negative hat or negative X direction, so we'll just call it. I have plums. I didn't mean to make my parentheses looks so big. That's more reasonable. So plus the white component of B one. So this is gonna be the one times the co sign of 45 degrees, and this is going to determine our J hat or why direction for our B one better. Okay, so now we need to add on to that our X and Y components for B two. So be too if you go back and look at the figure. It's pretty easy to see from the figure that B two on Lee has X components, right? The red vector. They're only has X components and know why. Components so be too. Is just gonna be two in the negative. If you look, it's the negative ex direction. So we're defining negative as to the left And in this figure, both B one and B to point to the left. So negative x direction. OK, so carrying this out on Lee, add like components of the eye hat and the J hat components get at it together. So we have negative 6.51 times 10 to the minus six. Tesla in the eye had direction for being met, plus 1.51 times 10 to the minus six. Tesla in the J had direction. So this technically is R B of X value, and this is our viel I value. Okay, so then, to find the magnitude of the net, call this magnitude of the vector. We can use Pythagorean theorem, which says that it's just the X component of the net squared, plus the white component of being that squared. So plugged those values. And when we find that this magnitude is equal to 6.68 times 10 to the minus six Tesla And lastly, we need to find direction, box set and as our magnitude. And we also need to find direction for part B. So the directionality the angle will call the state, uh, is going to be equal to using our econometric identities. This is the inverse tangent of being Why, Overby of X. This is because the tangent of fate owns B A y over B of exodus solve for theta. We take inverse tangent of be awhile, Kirby of X playing those values, and we find this is 33 degrees, Okay, but 33 degrees, um, above the negative x access. So we need to We need to make sure we define 32 3 degrees relative toe. What? It's not relative to the positive X access. This is 33 degrees above the negative X axis. What's boxes him is the final part of the solution to our question.

This question, we're talking about magnetic fuel created by current. So in this situation we have uh two wires carrying current and the direction is out of the screen. Um And I is equal to four uh amps. And whereas to find what the magnetic field is at point P one, P. Two and P. Three. Um so the distance between the two parents is 10 centimeters and P one is directly in the middle of the two currents or that she wires. And then P. Two is 25 centimeters to the right of uh this wire, and then P. Three is 20 centimeters above the one mm. So question A. Asked what the the magnetic field is at 0.1 very few one.

To current getting better wires. Here is the A wire. And here's to be wire. The current travels in the same direction. They are 10 centimeter apart and in part off the problem we are asked to find to find a B, uh, medic field due to the while end. Be at the point people in here in the midway between and B you want to find it, we find the maybe feel due to ah, what a And if you feel, do tow be. And we had bought off them to find the net. Man tick field you to A and B, This is net field. So let's find a B a b a We confined it. Ah, mule. I divided by two pi r and we get the dismantle field to be 16 times 10 to the power minus six. Tesla, that is up in our direction. Excuse me? A man different due to poor s o b will be knew. I you bye bye to buy our That is, uh, 16 16 times 10 to the power minus six. Tesla. That's insane. Made you because off the same guard that's in a downward. Um then because the directions are opposite. So they're so this is up. And plus this is down. They will cancel oat. And we get the net romantic feel to be a zero and point B one for a part to be, um this is again a This is a B. We are asked to find at the point p two, what is the Mayfield due to end B A distance or for 25 centimeter from the midpoint between envy huges tens? And did you do here again? I will use the sand procedure find maybe 50 to a and B and head up. So maybe feel due to a he's substituting your values for by times 10 to the power minus seven. Ah, times four here, divided by to buy. Does five plus 25. This is this distance and here this distance. So this would be five for a wife in plus 25. Um, this will be 2.66 times 10 to the power minus six. Tesler, this is in direction. Up and for But, uh, for for the B. What? This will be a full by time. Stepped up our minor seven times four. This will be 25 minus life. We need only dis distance from here to here, which is a 20 centimeter. This is a two by times 20. And this gives us be Tesla G Oh, to be four times 10 to the power minus six. Tesla, that is again up. So we led both these together. So we will be then. B A plus B B, this is 6166 times 10 to the power minus six. That's what This is an up direction for part C. But she we had asked to find again. This is a A and this is B. We were asked to find it a point three s. Oh, Petey. Here, Petey, Find a medic. Three little Petri. Here is the P one. And here it was a tree, too. And the distant work. The list is up to be three. He's 20 centimeter. And for me and a year is a five cent in nature. Well, first, find a b A and be a We can find that my, um went applying four pi times 10 to the power minus seven times four, divided by distance, which is a two by times five square root off her 17 times 10 to the power of minus two. We find this distance by taking using a right triangle that is a 30 here this distance we know it's a five centimeter And what, Nicholas twenties and the richer so this distance will get is a five square root over 17. And this is our our for a while for the B one and the direction we can get by using I I crows are and the direction we get this four I cross on will be well minus 20. Um 54 square root off 17. This is an I A plus five over five square root over 17. This is in J where Z's, you know, here for, uh, b Manti tree due to the B and B again four pi times 10 to the power minus seven times four divided by for to buy times five square root off by using a symmetry, we can just write five squared off 17 times sent to the ball minus two is the distance. This gives us 3.88 times 10 to the minus six Tesla and again we can find the direction by Crossman declined. I with the unit director are that is Ah, minus 20. Divided by five square. Root off 17. Here, this is an I in direction and plus minus five five. Divided by five. Scared or 17. This is InGen erection and Z's. You know, here. Then we can find the net. May dude off. Um, net be will be be a plus, baby. And this is 3.8 times 10 to the power minus six. And it's a director or minus 20. Derived by five square root off 17. This is an eye direction plus five Do a bay five. 17. This is in J direction. Where Z zero plus 3.88 times 10 to the power minus six. Minus 22 ever. 5 17 Square it or 17. This is and I minus five. Do you have a five square Google for 17? This is in jail direction where Z direction is zero. This will give us the tree, uh, 7.53 times 10 to the power minus six. This is toward us. Worse, the left be our let we will be No


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