3. Consider the square loop of wire with a curreat running in it shown in Figure

Question

3. Consider the square loop of wire with a curreat running in it shown in Figure 1. The loop lies in the xy-plane centered on the origin. The current running in the wire is 1.0 A and it runs counterclockwise when the wire is viewed looking down the z-axis. The loop has a side length of 2.0 cm. (a) Consider the Amperian path, A, (shown in the figure) that is a square lying in the xy-plane, centered on the origin with side length 4.0 cm. What is the line integral of the B-field going around this path counterclockwise as viewed looking down the z-axis? (b) Consider the Amperian path, B, (also shown in the figure) that is a square lying in the yz-plane, centered on the origin, with a side length 4.0 em. What is the line integral of the B-field going around this path counterclockwise as viewed looking back along the x-axis? (e) Consider the Amperian path, C, (also shown in the figure) that is a square lying in the yz-plane, centered on the point (1.0, 0, 0) cm, with a side length 20 em. What is the line integral of the B-field going around this path counterclockwise as viewed looking back along the x-axis (d) Can any of these Amperian paths be used to calculate the B-field strength at some location? If not, explain why not. If so, caleulate the B-field magnitude at that location. Figure 1: A square current loop in the xy-plane and 3 Amperian paths.

Explanation / Answer

According to Ampere's circuital law, the line integral of magnetic field over a loop is equal to uoInet where refers to the permeability of free space and Inet is the net current flowing perpendicular to the coil.

a. As this loop lies in the plane of the current loop, no current is cutting this loop. Hence Inet is zero and line integral of B-field is zero.

b. Out of the 4 sections of the square loop, 2 sections are along the loop B and the remaining two cut the loop B in opposite directions cancelling each others effect such that again Inet = 0 and hence line integral of B-field is zero.

c. Here, the loop C intersects with the current section at two instances. In such a situation Ampere's circuital rule cannot be applied. (This is similar to the fact that Gaussion surface cannot pass through a discrete charge). Also, the current element is neither inside and nor outside the loop.

Hence,

No amperian path can be used to find out the magnetic field at any point in this situation for the reasons stated above.

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