Massive hints for your imminent HW set:
1. If you can figure out the field due to a spherical distribution of charge (everywhere, inside and outside of the sphere), you can integrate the field squared over volume elements making up all space to get the energy. Use spherical shells of volume 4(pi)r^2, and integrate from radius 0 to infinity. The field outside is just that of a point charge (charge density times volume ...), the field inside is a point charge but the charge is how much is contained in a radius r from the origin.
2. Trivial once you've done the first problem.
3. We'll do this in class Wednesday, for the most part.
4. This one is hard, and just to be a bastard I'm going to let you think about it for a bit :-) [Further hints will follow though.] First you'll need the capacitance of a spherical capacitor; this is a standard result (i.e., google) if you don't see how to do it. We'll touch on this in Wednesday's class. The energy of the capacitor is U=Q^2/2C, where C is the capacitance and Q the charge stored. Write Q in terms of the electric field at the surface of the inner sphere (it behaves as a point charge outside!). Differentiate that with respect to the inner sphere radius to maximize, plug the value of the inner radius obtained back into the U equation.
5. Did this in Monday's class ...
6. Superposition: two lines plus a semicircle. The lines segments both give the same contribution to the potential; find one and double it. Any bit dx of the line has charge (lambda)dx, giving a potential dV=k(lambda)dx/x. Integrate from x=R to x=3R. The semicircle should be easy enough ... every bit of it is the same distance from the origin.
7. We did most of this in class; sum potential energies for all pairs.
8. We'll do this in class Wednesday.
1. If you can figure out the field due to a spherical distribution of charge (everywhere, inside and outside of the sphere), you can integrate the field squared over volume elements making up all space to get the energy. Use spherical shells of volume 4(pi)r^2, and integrate from radius 0 to infinity. The field outside is just that of a point charge (charge density times volume ...), the field inside is a point charge but the charge is how much is contained in a radius r from the origin.
2. Trivial once you've done the first problem.
3. We'll do this in class Wednesday, for the most part.
4. This one is hard, and just to be a bastard I'm going to let you think about it for a bit :-) [Further hints will follow though.] First you'll need the capacitance of a spherical capacitor; this is a standard result (i.e., google) if you don't see how to do it. We'll touch on this in Wednesday's class. The energy of the capacitor is U=Q^2/2C, where C is the capacitance and Q the charge stored. Write Q in terms of the electric field at the surface of the inner sphere (it behaves as a point charge outside!). Differentiate that with respect to the inner sphere radius to maximize, plug the value of the inner radius obtained back into the U equation.
5. Did this in Monday's class ...
6. Superposition: two lines plus a semicircle. The lines segments both give the same contribution to the potential; find one and double it. Any bit dx of the line has charge (lambda)dx, giving a potential dV=k(lambda)dx/x. Integrate from x=R to x=3R. The semicircle should be easy enough ... every bit of it is the same distance from the origin.
7. We did most of this in class; sum potential energies for all pairs.
8. We'll do this in class Wednesday.
No comments:
Post a Comment