3. Wire and Circular Loop Consider a very long straight wire that lies on the ar-axis. The wire carries current. 11 in the +m-direction. The magnetic eld due to the wire at any point in the arty-plane is _ LOIL 21m A circular loop of wire has a radius R and lies in the ' .ry-plane with its center at (0 2R 0). It c'ar ries a countercrlotkwise current 12 (viewed from a point on the _'.+z-axis) We dwant to nd the net magnetic force on the loop due to the straight wire using F: I I2 d? x B (a) (d) _) F ind an expression for (if, an innitesimal displacement around the circular loop in the direction of the current, in terms of 12,6, (i9, and unit vectors i,j,k, as needed. The angle 0 is defined in the gure above. Consider the direction of the force on an innitesimal segment, dB : I2 df'x B. Based on the properties of the cross-pro duct and symmetry, identify which two of the three Cartesian components of B have to be zero and explain the reasoning behind this conclusion carefully. What is the expression for (if; which may show the angle (6') dependence? Now set up your integral Ii" 2 f I2 df'x B, simplifying the integrand as much as possible. Be sure to write the limits on the integral. Perform the integration: what. is the net magnetic force on the loop? Feel free to use Wolfram Alpha or another such tool to help perform the integration. Imagine that the ring with the current 12 is moving toward or away from the innite wire with the current 11 because of the magnetic force you found in part (c). The kinetic energy should increase with time. Does this mean that the energy conservation law is violated? If not, where does this increase in kinetic energy come from? This is more like a conceptual question with open discussion, so you don't have to derive the kinetic energy