Final Questions (Take Home) For a stack of N concentric circular loops, each of radius "a" and carrying current IH (see figure below), the B field along the x-axis shown is given by eqn (5): В - 2(x +a) H (5) 2 3/2 Q1.) One stack: For one such stack, make a rough sketch of the shape B vs. x. Include both + and - x values. (Use your graphing calculator or Excel to expedite-use any non-zero values you like for the constants.) With only visual inspection, identify the one point of zero slope and at least one point of inflection (there is one on either side of x-0.) Label these on your graph. One stack: Two stacks: N a a a X0 +X +X X+L x+L/2 Q2.) Two stacks (general): Suppose you had two identical stacks, located at x-0 and x-+L respectively. Using the ideas of Q1, you can see what Bnet VS. X looks like for this double stack, once you have some idea of how large ""L" is. a.) First, explain conceptually why the overall slope must be zero at x-+L/2 (the midpoint between the stacks) regardless of the separation distance L. isn Then sketch the individual B vs. x from each stack (dotted line) and then Bnet VS. X (solid line) for the cases described below. (Make sure your graphs are still in agreement with part a!) b.) assuming a large L value. (Both stacks contribute the concavity seen at large distances in Q1.) c.) assuming a small L value. (Both stacks contribute the concavity seen at small distances in Q1.) Q3.) Two stack (special case): Our set-up is called a "Helmholtz coil". This isa two stack system, with the separation L chosen specifically so that the midpoint of the stack also coincides with the inflection point of both individual fields. a.) At this midpoint, what is the slope of Bnet? How about the concavity? b.) What is the whole purpose of the Helmholtz coil being constructed this way? To make this happen, one must choose L-a for the Helmholtz coil (stack separation loop radius.) c.) What is then the x-value at the midpoint? Using this and eqn (5), show that the B field on the x axis halfway between the two stacks is indeed given by eqn (4). Use symmetry arguments to expedite. d.) By hand, show this x value is indeed the inflection point of eqn 5. CHECK e.)(BONUS) Derive eqn (5) starting from the Biot-Savart Law. (lf this has not yet been covered in lecture, ask your instructor if you can hand this in later on.) .0eNidoeig ox 128 Lab # 11: Electron charge-to-mass ratio - p. 8 ит е