Find an expression for the magnetic field strength at the center ...

Question

Find an expression for the magnetic field strength at the center (point P) of the circular arc in FIGURE P29.45.

Accepted Answer

Hello everyone. Let's go through this practice problem. The figure shows an electric current flowing through a wire with two straight segments and a curved portion, calculate the magnetic field at 0.0. So we're given a diagram of the situation and we're given four multiple choice options. Option, a 0.16 micro teslas, option D 3.5 micro teslas option C 1.6 milli teslas and option D 35 milli teslas. So for problems like this where we're analyzing the magnetic field generated by a current when it's an elaborate shape like this, we generally want to apply the Bo Sarla which states that the magnetic field differential is equal to the magnetic constant mu knot divided by four pi multiplied by the current through the wire multiplied by the length of differential crossed with the unit vector pointing from the current element to the field point known as our hat divided by R squared where R is the distance between the current element and the field point. This is often a pretty big and scary looking equation. But there are a lot of ways we can simplify it and we can simplify it pretty nicely for this problem as well. First off, for the two stray segments of the wire note that those segments of wire are parallel to our hat parallel to the unit vector that would stretch from the point and the wire itself. So that means that for those sections, for those segments, DL is parallel to R hat. And since DL is being crossed with R hat, the terms go to zero for those segments. So they can be ignored as they won't contribute any magnetic field. For the curved segment of the wire. Notice that the wire itself is always going to be perpendicular to any point stemming from 0.0. So again, because of the way cross products work, this means that that magnitude will actually be maximized. So for that curved portion DL crossed with R is going to be equal to just DL because it's being maximized. So our more simplified form of the B OS our law for this problem can be written as Muno divided by four pi multiplied by the current divided by R squared, which in this case can just be thought of as big R, the radius of the se of the curved segment due to the fact that that's all the distance we're considering. So big R squared multiplied by DL, the differential length of the wire. The problem is asking us to find that magnetic field. So to find B will integrate BB. So B is of course, equal to the integral of the differential element. So that means that B is equal to mu knot divided by four pi multiplied by the current divided by the square of the radius multiplied by the integral over the length of the curved portion of the wire. The diagram tells us that we're integrating over an angle of theta equals pi divided by four radium. So let's set our bounds of integration to be from zero to theta equals pi divided by four. And then let's rewrite our DL in terms of theta. Recall that the length of a curve is equal to the radius of the curve multiplied by the angle subtended by that curve. So DL is equal to R, the radius of the curve multiplied by the angle theta which in our case is going to be a theta element because that's what we're integrating since R is constant though one of them can be canceled out, simplifying this down even further. This ends up turning into a very simple integral because the integral of D theta is just gonna be theta. So we just multiply the whole thing on the outside of the integral by theta. So our final formula for the magnetic field is Muno divided by four pi all multiplied by I theta divided by R. So all that's up for us to do is plug in the values that were given to us by the problem. So the magnetic constant is equal to four pi multiplied by 10 to the power of negative seven tesla meters per amp here multiplied by the current value which is given in a diagram as one amp here multiplied by the theta value which again was given in the diagram as pi divided by four radians, all divided by four pi multiplied by the radius of the arc which is given in the diagram as 0.5 m. So if we put all of this into a calculator, then we find a magnetic field of about 1.6 multiplied by 10 to the power of a negative seven teslas, which can alternatively be written as 0.16 micro teslas. And so that is the answer to our problem, which if you look at our multiple choice options corresponds to option A. So a is the correct answer to this problem. And that's all for this video. I hope this video helped you out. And if it did, please consider checking out our other tutoring videos, which will give you more experience with these types of problems. That's all for now. And I hope you all have a lovely day. Bye bye.

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Find an expression for the magnetic field strength at the center (point P) of the circular arc in FIGURE P29.45.

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