Question

1. This quasi-"walkthrough" problem is great practice in cross-products, vectors, and integration. Consider the current loop shown in Figure 2, with B _Bx, and the loop lying in the x - z plane of the page (y points into the page). We wish to find the net torque on this current loop. we'll do this by integrating in θ, the angle shown (a) We'll start with a little segment dl at point p as shown in the figure. What is the magnitude dl of dl, in terms of r and dθ? (b) Write the vector r, which points from the origin to P, as a sum of an x- and z-component, e.g. where you fill in the (...) parts. high-v ow-V FIG. 1: For problem II.3 (c) Now do the same for dl. Verify that your expression makes sense by evaluating it at θ-0 (you should get d12) and at θ π/2 (you should get-d12) (d) Now compute dF-idl x B. You should find something in the -y direction that depends on θ and has the units of force (e) Now compute dT-X dP. You should get two terms, one in the-z direction and one in the +x direction, one of which has a cos2 θ in it and one of which as a cos θ sin θ in it (f) Now plug this in to the integral You can split this into two parts. The one with cos θ sin θ is zero (why?) The other uses extra credit: can you give a simple explanation for this integral? (g) Evaluate to show that you get the correct result Whew! Now you can see why I skipped going through this in lecture!

Answer 1

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