The line integral is the same for all three loops shown below. Loops 1 and 2 hav

Question

The line integral is the same for all three loops shown below. Loops 1 and 2 have the same perimeter. Consider the following discussion about the incorrect predictions made by three students: Student1: "It seems like the integral around Loop 2 should be greater. They're the same length, but in Loop 2 the magnetic field always points along the path, and in Loop 1 only a component of the magnetic field points along the path." Student2: "I thought that Loop 3 should be the largest integral, since the loop is closest to the wire so the magnetic field is strongest." Student3: "I predicited the opposite. The path is shorter around Loop 3, so the integral around Loop 3 should be less than that around Loop 2."

Explanation / Answer

See in all the cases the value of b.dl would be same as by the ampere circuital states tha surface integral b.dl=uoi. The first student is saying the wrong statement as the surface integral b.dl will not change if u take a square as the length would be different for path and the sum of dot product would make it equal. The second student is also wrong as we approach nearer to the wire the length becomes smaller but the value b.l remain same. The third is wrong as B.dl would be same and equal to uo.i

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