Partnership for Integration of Computation into Undergraduate Physics

####Exercise 1 An external applied magnetic field (uniform over space, sinusoidal in time) exists perpendicular to the plane of a single circular loop of wire. Create a program that asks a user for the field amplitude (T), frequency (Hz), loop radius (m), and loop resistance (ohms). The program should then output the induced magnetic field amplitude (T) in the center of the loop. The following shows a step-by-step solution that should be attempted by hand before programming. * applied magnetic field: $B_\mathrm{{ext}}(t)=B_0 \mathrm{sin}(2\pi ft)$ * angular frequency: $w = 2\pi f$ * magnetic flux: $\Phi_B= \int \vec B\bullet d\vec A = B_0\pi r^2\mathrm{sin}(\omega t)$ * induced emf: $\mathcal{E}(t) = \displaystyle-\frac{d\Phi_B}{dt} = -B_0\pi r^{2}\omega \mathrm{cos}(\omega t) = -\mathcal{E_\mathrm{max}}\mathrm{cos}(\omega t)$ * induced current amplitude: $I_\mathrm{{max}} = \displaystyle\frac{\mathcal{E}_\mathrm{{max}}}{R}$ * induced current: $I(t)=-I_\mathrm{{max}}\mathrm{cos}(\omega t)$ * induced magnetic field amplitude in the center of the loop (via the Biot-Savart Law): $B_\mathrm{{max}} = \displaystyle\frac{\mu_0I_\mathrm{{max}}}{2r}\rightarrow B_\mathrm{{ind}}(t)=-B_\mathrm{{max}}\mathrm{cos}(\omega t)$ With $B_0$ = 100 mT, $f$ = 60 Hz, $r$ = 5 cm, and $R$ = 0.5 $\Omega$, this gives $B_\mathrm{{max}} = 7.44\times10^{-6}$ T in the center of the loop. ####Exercise 2 Display the directions of 1) the external magnetic field, 2) the induced magnetic field, and 3) the current in the loop over time. Be sure to distinguish the arrows with labels. The directions of the current and magnetic field are oriented in accordance with Lenz's law as described above. ####Exercise 3 Answer the following conceptual questions through calculations (e.g., by hand or spreadsheet) and/or programming. * What is true about the external magnetic field (and its time rate of change) when the induced magnetic field is at an extreme, and vice versa? How do you know visually, and how is it seen in the relevant equations for each? * Mathematically, what causes the external and induced magnetic fields to be 90° out of phase? Would this happen for any function of the external field, or is it specific to this particular function? Similarly, what causes the induced current and induced magnetic field to be in phase, and is that function-dependent? * If the magnitude of the external magnetic field were linearly increasing in the upward direction, how would the induced magnetic field behave (in both magnitude and direction)? * If the magnitude of the external magnetic field were exponentially increasing in the upward direction, how would the induced magnetic field behave (in both magnitude and direction)? * If the external magnetic field were constant (and non-zero), how would you expect the induced magnetic field to behave? What if the ring were moving or began to tilt? * What happens to the induced magnetic field when the frequency is increased? Why?