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Find the magnitude and direction of the magnetic field at the point equidistant from the wires in Figure $22.64(b),$ using the rules of vector addition to sum the contributions from each wire.

$B=1.06 \cdot 10^{-5} \mathrm{T}$

Physics 102 Electricity and Magnetism

Chapter 22

Magnetism

Magnetic Field and Magnetic Forces

Sources of Magnetic field

Electromagnetic Induction

Inductance

Simon Fraser University

University of Winnipeg

McMaster University

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here. We know that this would be the diagram of the system. And we know that the magnetic field is equaling the mag native permeability in a vacuum multiplied by the current divided by two pi times are are being the shortest distance to that wire. This would be the magnetic field produced by a long straight wire. And so from this diagram, we can first say that the distance D is going to be equaling to the square root of 0.10 meters quantity squared plus again 0.10 meters quantity squared. And this distance is equally one. Rather, we can say 14 points. Actually, you simply say 0.1414 meters. Now we know that here four part fort Rather, we could say the magnetic permeability sub one. This would be according to eye sub one. Now, you can actually say here that this would be equaling two. The same thing that we're using this equation here and we're essentially going to say for pie times 10 to the negative seventh test the meters per amps. This would be multiplied by here. 10 amps divided by two pi multiplied by our 0.1414 meters and we find that beast of one is equaling 1.4 14 times 10 to the negative fifth. Tesla's Keep track of all these values. Of course. Here, we're going to then say that be sub three. In this case, this is the magnetic field at oh due to wire C. And so we can say that then this would be equaling four pi times 10 to the negative seventh Tesla's meters per amps multiplied by five amps. Again, this is do the wire C. This would be divided by two pi multiplied by the shortest distance, again being 20.1414 meters. And we find that the magnetic field at oh due to wire C is equaling 7.7 times, 10 to the negative sixth. It might be easier to write 0.707 times 10 to the negative fit test list and from symmetry, we know that these up to is equaling beasts of one where be sub four is equal and be sub three. And so the magnetic field on the X axis would be equaling piece of one plus piece of two plus piece of three plus piece of four multiplied by co sign of 45 degrees. And we can say that to be sub X is gonna be equaling 1.414 times, 10 to the negative fifth plus 1.414 times, 10 to the negative fifth plus 0.707 times 10 to the negative fifth plus 0.707 times, 10 to the negative fifth. This would, of course, be for Tesla's and then Times Co sign of 45 degrees. And we find that the magnetic field on the X axis is equaling 3.0 times, 10 to the negative fifth test floods and the Y direction. By symmetry, this would be equaling. Be sub one minus b sub two minus B said three plus be sub four. And you know that this would be times sign 45 degrees. But however you know that all these you're gonna cancel out and this would equal zero Tesla's so here we can simply say that the magnitude of the magnetic field is equaling 3.0 times, 10 to negative fifth Tesla's and, of course, Fada with the equaling zero degrees because this entire magnitude is in the ex direction. This would be our final answer. That is the end of the solution. Thank you for one.

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