Question

2. A steady, uniform magnetic field B with unknown magnitude and direction exists through- the shaded rectangular region shown here (diagrams are not to scale). This region is planar-entirely in the x-y plane (+x is east and +y is north). The region's width (from its west edge to its east edge) is a known distance D. The region's length (from its north edge to its south edge) is 10D. Consider the following experiments, all performed with this field. coonfinate atan north out easr Experiment : A single circular loop of wire of diameter D is given a net charge of +Q, distributed uniformly along the wire. The loop is then placed with its center located at the exact center of the field, as shown here (view of D B the loop is edge-on). Then the loop is made to spin-like a wheel-around a north-south axis through its center, at a known steady rate o (given in rpm) with the tangential velocity vectors for its westernmost and easternmost points as shown. Result: The total magnetic field at the loop's center is zero. Experiment (ii): The loop's spinning is halted, and the loop is discharged; it's now a neutral wire. Then the wire is cut and re-shaped into a single straight length, and held randomly within the field (so that its entire length is in the x-y plane and completely within the field). A steady current / is then sent through it and the magnetic force acting on the wire is measured. Using that same current , the wire is re-positioned various times (always holding it completely within the field) until the maximum possible magnetic force magnitude has been achieved. Call this value F mag max Experiment (ii): Now the wire is rotated so that its current (still the steady / from experiment ii) is flowing directly South. Then a second identical length of wire is fixed in place parallel to the first wire, at a distance D/50 to the east of the first wire. This second wire carries three times as much current as the first wire, and its current also flows south Find an expression for the total magnetic force (magnitude and direction) exerted on the first wire. a. b. Find an expression for the total magnetic field magnitude at a point midway between the two wires Your answers should be expressions that include only known values, and for full credit, you must show that the units of your solutions are also correct. These values are known: D, Q,oF mag max

Answer 1

"Experiment (i): lambda = Q/ pi D Angular velocity W = W_rpm rpm = 2 pi w_rpm/ 60 rad/ s Direction of W_rpm is along + y axis (North) Equivalent current I = Q/ T = Q/ 2 pi 2 pi W_rpm/ (60) I = Q W_rpm/ 60 Magnetic field at the center of loop due to notation of wire loop B prime = mu_0 I/ 2r = mu_0 I/ D = mu_0 theta W_rpm/ 60 D By right hand rule, B prime^bar is along North (+ y) B_total^bar = 0, B^bar + B prime^bar = 0 = > B = mu_0 Q W_rpm/ 60 D with direction of B^bar along - y (South) Experiment (ii): Length of wire l = pi D F^bar = I l^bar times B^bar,