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HW21_pdf - mcconnell(kam2342 – homework21 – Turner...

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mcconnell (kam2342) – homework21 – Turner – (60230) 1 This print-out should have 12 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 (part 1 of 2) 10.0 points A current I = 3 A flows through a wire perpendicular to the paper and towards the reader at A and back in the opposite direc- tion at C . Consider the wires below the plane at A and C to be semi-infinite. In the figure, L 1 = 5 m, R = 4 m, and L 2 = 7 m and there is a B = 9 . 89 T magnetic field into the paper (not including the field due to the current in the wire). Caution: It may be necessary to take into account the contribution from the long straight wire which runs up to and down from the underneath side of the page. 5 m 3 A 3 A 7 m 3 A 4 m A C O B = 9 . 89 T What is the magnitude of the force on the wire due to the external magnetic field B ? Correct answer: 421 . 69 N. Explanation: Let : R = 4 m , I = 3 A , L 1 = 5 m , L 2 = 7 m , and B = 9 . 89 T . By the Biot-Savart law, d vector B = μ 0 4 π I dvectors × ˆ r r 2 . The contribution from the long straight wire which runs into and out of the page is zero since the external field and the current are parallel. The force on a current carrying wire from point A, at vectorr 1 , to point C, at vector r 2 , in a uniform field is vector F = I integraldisplay vectorr 2 vectorr 1 ( dvectors × vector B ) . Since vector B is a constant, it can be taken out of the integral and we can write (recalling that if we change the order of the cross product, we need to change the overall sign) vector F = I vector B × integraldisplay vectorr 2 vectorr 1 dvectors = I vector B × ( vectorr 2 vectorr 1 ) . Now vectorr 2 = ( R + L 2 ) ˆ i vectorr 1 = ( R + L 1 ) ˆ j vector B = B ˆ k . Thus, the total force is vector F = I B [( R + L 2 ) ( ˆ k × ˆ i ) ( R + L 1 ) ( ˆ k × ˆ j )] = I B [( R + L 2 ) j + ( R + L 1 ) i ] . Thus the magnitude of the force is F = I B radicalBig ( R + L 2 ) 2 + ( R + L 1 ) 2 = 421 . 69 N . 002 (part 2 of 2) 10.0 points What is the magnitude of the magnetic field at the center of the arc O due to the current in the wire? Correct answer: 1 . 25435 × 10 7 T. Explanation: The two straight current segments within the plane of the paper do not contribute to the magnetic field at point O , because they are parallel to the radius vector from that point. Therefore dvectors × ˆ r = 0 on these segments. The
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mcconnell (kam2342) – homework21 – Turner – (60230) 2 contribution from the curved part of the wire is easy to find using the Biot-Savart law B = integraldisplay μ 0 I dvectors × ˆ r 4 π r 2 = μ 0 I parenleftBig π 2 parenrightBig 4 π R = μ 0 I 8 R ( ˆ k ) , but one must not forget to take into account the contribution from the long straight wire which runs into and out of the page. For the wire above this is B 1 = μ 0 I 4 π ( L 1 + R ) ˆ ı , while for the wire below we have B 2 = μ 0 I 4 π ( L 2 + R ) ˆ  .
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