The magnetic field at a due to the upward current is

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The magnetic field at A due to the upward current is B up,A = µ 0 (2 I 0 ) 2 π ( r/ 2) = 2 µ 0 I 0 π r . The right-hand rule tells us the direction is into the paper. Due to the fact that A is the same distance from both wires, the total magnetic field at A is B A = 2 µ 0 I 0 π r + µ 0 I 0 π r = 3 µ 0 I 0 π r . Now, the field at B due to the upward current is B up,B = 2 µ 0 I 0 2 π (3 r/ 2) = 2 µ 0 I 0 3 π r again into the paper, while the downward current gives B down,B = µ 0 I 0 2 π ( r/ 2) = µ 0 I 0 π r out of the paper. So at B, the net field out of the paper is: B B = B down,B B up,B = µ 0 I 0 π r parenleftbigg 1 2 3 parenrightbigg = µ 0 I 0 3 π r .
mittal (im5936) – Magnetic Force and Field HW – yeazell – (58010) 9 Comparing their magnitudes, we find B A B B = 3 µ 0 I 0 π r µ 0 I 3 π r = 9 . 020 10.0points Consider a long wire and a rectangular current loop. A B C D I 1 b a I 2 Determine the magnitude and direction of the net magnetic force exerted on the rectan- gular current loop due to the current I 1 in the long straight wire above the loop. In order to use this, we need to know the magnitude and direction of the magnetic field at each point on the wire loop. We can apply the Biot-Savart Law. The result of this is that the magnitude of the magnetic field due to the straight wire is B = µ 0 I 1 2 π r , and the direction of the magnetic field is given by the right hand rule; the field curls around the straight wire with the field coming out of the page above the wire and the field going into the page below the wire. We can now find the force on the segment AB ; applying the right hand rule to find the direction of the cross product, dvectors × vector B , we see that the force will be in the up direction. Since the wire along the segment AB is straight and always at a right angle to vector B , the cross product simplifies to B ds . Since the magnitude of the magnetic field is constant along segment AB , it can come out of the integral which simplifies to give us the result,
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