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chp30_post - Chapter 30 Problem 10 A single piece of wire...

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Chapter 30 Problem 10 A single piece of wire is bent so that it includes a circular loop of radius a, as shown in Fig. 30-48. A current I flows in the direction shown. Find an expression for the magnetic field at the center of the loop. To solve this problem you will either need to use the equations for the magnetic field of a current loop and a long straight wire, or you will need to derive those equations from the Biot-Savart law. Using the right hand rule, you can see that the magnetic field from the loop and from the wire will both be out of the page. The equations for a loop and for a long wire are given by 30-3 and 30-5 respectively. 2 / 3 2 2 2 0 ) ( 2 a x Ia B loop + = µ is the equation for a loop and is derived on page 773. x is the distance from the center of the loop, which is just 0 in our case because the loop is in the plane of the page. a is the radius of the loop. y I B line π µ 2 0 = is the equation for a line and is derived on page 776. y is the distance from the wire which in our case also happens to be a, the radius of the loop. Since both fields point out of the page, we can add them as scalars. a I B a I a I B a I a Ia B B B B total total total line loop total π π µ π µ µ π µ µ 2 ) 1 ( 2 2 2 ) 0 ( 2 0 0 0 0 2 / 3 2 2 2 0 + = + = + + = + = Problem 15 Part of a long wire is bent into a semicircle of radius a, as shown in Fig. 30-50. A current I flows in the direction shown. Use the Biot-Savart law to find the magnetic field at the center of the semicircle (point P). The Biot-Savart law is given by × = 2 0 4 ˆ r r l Id B π µ v v dl is the vector along a given section of the current. The dl along the straight portion of

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chp30_post - Chapter 30 Problem 10 A single piece of wire...

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