- #1
E_M_C
- 43
- 0
The problem is stated:
The preceding problem was an artificial model for the charging capacitor, designed to avoid complications associated with the current spreading out over the surface of the plates. For a more realistic model, imagine thin wires that connect to the centers of the plates (the plates are circular). Again, the current I is constant, the radius of the capacitor is a, and the separation of the plates is w « a. Assume that the current flows out over the plates in such a way that the surface charge is uniform, at any given time, and is zero at t = 0.
Find the electric field between the plates, as a function of t.
I understand that I have to use Gauss' Law to find the E-field, but first I have to find the charge distribution Q(t), this is where I'm having some difficulty. After a lot of frustration, I peeked at part of the solution, and I found that the charge distribution is Q(t) = It. I assume it was arrived at as follows:
[itex] I = \frac{dQ(t)}{dt} → \int dQ = \int I dt → Q(t) = It[/itex]
I was surprised to see that the charge distribution is a linear function, as I was expecting an exponential expression. Maybe I'm not getting the concept of a "uniform surface charge", but I don't see how a linear charge distribution is a "realistic model." For example, what happens as t → ∞? The charge distribution blows up. How is that realistic?
Any help is appreciated.
The preceding problem was an artificial model for the charging capacitor, designed to avoid complications associated with the current spreading out over the surface of the plates. For a more realistic model, imagine thin wires that connect to the centers of the plates (the plates are circular). Again, the current I is constant, the radius of the capacitor is a, and the separation of the plates is w « a. Assume that the current flows out over the plates in such a way that the surface charge is uniform, at any given time, and is zero at t = 0.
Find the electric field between the plates, as a function of t.
I understand that I have to use Gauss' Law to find the E-field, but first I have to find the charge distribution Q(t), this is where I'm having some difficulty. After a lot of frustration, I peeked at part of the solution, and I found that the charge distribution is Q(t) = It. I assume it was arrived at as follows:
[itex] I = \frac{dQ(t)}{dt} → \int dQ = \int I dt → Q(t) = It[/itex]
I was surprised to see that the charge distribution is a linear function, as I was expecting an exponential expression. Maybe I'm not getting the concept of a "uniform surface charge", but I don't see how a linear charge distribution is a "realistic model." For example, what happens as t → ∞? The charge distribution blows up. How is that realistic?
Any help is appreciated.