Two very long parallel conducting strips, each of width a, have vertical separation b. The strips have negligible thickness, and they carry uniform currents of magnitude I in opposite directions. The object of this problem is to find the magnetic force per length on each strip due to the other.

(a) (2 points) What is the direction of the net force per length on the lower strip? Briefly explain your reasoning. Let's place the lower strip in the xy-plane with its midline along the y-axis. To begin let's calculate the magnetic field due to the upper strip at the point P = (x, 0, 0). We can divide the upper strip into parallel flat "wires" of width dr' and infinite length, then sum their net effect at the field point. (b) (1 point) What is the current dI' carried by each wire? Express your answer in terms of dr', I and a. (c) (3 points) Suppose a wire crosses over the x-axis at . Make a sketch with dI' pointing out of the page that shows the relationship between the wire and point P. Use the right-hand rule to draw in the magnetic field dB due to the wire at P. (d) (3 points) What are dB, and dB, due to the wire at P? Your answer should be expressed in terms of po, I, a, b, r, r' and da'. (e) (3 points) Now find B, and B₂ at P by summing (i.e., integrating) the contributions in part (d). Note: We don't actually need B, for the force/length calculation, but let's find it anyway. (f) (1 point) Now let's divide the lower strip into flat wires of width dr. What is the current de carried by each wire? (g) (2 points) Consider the wire that crosses the x-axis at point P. Make a sketch with dI pointing into the page that shows the two components of the magnetic field from part(e) and the resulting components dF/length and dF₂/length on the wire. (h) (2 points) Write down the expressions for dF/length and dFz/length in terms of I,a, B2, B₂ and dr. (i) (1 point) What is the value of F/length for the full strip? Hint: This is a quick one.Explain your reasoning. (j) (2 points) Now evaluate F₂/length for the full strip. Your answer should be in terms of μo, I, a and b. As a check: for I10 A, a = 0.05 m and b = 0.01 m, the magnitude is 8.4 x 10-4 N/m.