Q4. VECTOR POTENTIAL II A) Griffiths Fig 5.48 is a handy "triangle" summarizing the mathematical connections between J, A, and B (like Fig. 2.35) But there's a missing link, he has nothing for the left arrow from B to A. Note the equations defining A are very analogous to the basic Maxwell's equations for B: V.B = 0 VA = 0 V x B = uj VXA =B So A depends on B in the same way (mathematically) the B depends on J. (Think, Biot-Savart!) Use this idea to just write down a formula for A in terms of B to finish off that triangle. B) In lecture notes (and/or Griffiths Ex. 5.9) we found the B field everywhere R inside (and outside) an infinite solenoid (which you can think of as a cylinder with uniform surface current flowing azimuthally around it. See Griff. Fig 5.35. Use the basic idea from the previous part of this question to, therefore, quickly and easily just write down the vector potential A in a situation where B looks analogous to that, i.e. B = Cd(s R) , with C constant. (Sketch this A for us, B please) (You should be able to just "see" the answer, no nasty integral needed!) For you to discuss: What physical situation creates such a B field? (This is tricky - think about it!) Also, if I gave you some B field and asked for A, can you now think of an "analogue method", i.e. an experiment where you could let nature do this for you, instead of computing it? It's cool - think about what's going on here. You have a previously solved problem, where a given I led us to some B. Now we immediately know what the vector potential is in a very different physical situation, one where B happens to look like J did in that previous problem. Q4. VECTOR POTENTIAL II A) Griffiths Fig 5.48 is a handy "triangle" summarizing the mathematical connections between J, A, and B (like Fig. 2.35) But there's a missing link, he has nothing for the left arrow from B to A. Note the equations defining A are very analogous to the basic Maxwell's equations for B: V.B = 0 VA = 0 V x B = uj VXA =B So A depends on B in the same way (mathematically) the B depends on J. (Think, Biot-Savart!) Use this idea to just write down a formula for A in terms of B to finish off that triangle. B) In lecture notes (and/or Griffiths Ex. 5.9) we found the B field everywhere R inside (and outside) an infinite solenoid (which you can think of as a cylinder with uniform surface current flowing azimuthally around it. See Griff. Fig 5.35. Use the basic idea from the previous part of this question to, therefore, quickly and easily just write down the vector potential A in a situation where B looks analogous to that, i.e. B = Cd(s R) , with C constant. (Sketch this A for us, B please) (You should be able to just "see" the answer, no nasty integral needed!) For you to discuss: What physical situation creates such a B field? (This is tricky - think about it!) Also, if I gave you some B field and asked for A, can you now think of an "analogue method", i.e. an experiment where you could let nature do this for you, instead of computing it? It's cool - think about what's going on here. You have a previously solved problem, where a given I led us to some B. Now we immediately know what the vector potential is in a very different physical situation, one where B happens to look like J did in that previous