Answer Happy

VI. PROBLEM 6. TIME-DEPENDENT AMPERE'S LAW. Two metal spheres carrying charges +Q and -Q are placed at distance L from one another. They are connected with a wire with resistance and slowly discharge through it. a Demonstrate that, for an arbitrary Ampere's loop, it doesn't matter whether you choose the surface enclosed by the loop in a way that the current pierces it or not. Hints: 1. You don't need to actually calculate the magnetic field. You are only asked to prove that the results are equal. 2. Choose an Ampere's loop. Choose an arbitrary surface 1 that is pierced by the wire. Don't take the integrals. 3. Choose another surface 2 that doesn't include the wire. 4. Add and subtract surface 1. When you subtract the surface 1, you have a closed surface (1 + inverted 2). You can use Gauss' law to calculate the electric flux through it. 5. Notice that the current, by definition, is the charge per unit time: 1 = dQ 6. Compare the results. If they do not coincide, check the orientations of your surfaces. dt