Potential and current...

[Estrella]

Suppose one has a conducting sphere with radius A with current I1 flowing into the sphere and current I2 flowing out of the sphere such that I1>I2. How long would it take for the sphere to increase in potential by B in seconds?

 

[bobsmith1492]

No homework problems allowed. The whole point of homework is to figure things out by yourself.

 

[KIAman]

42

 

[PM650]

t = 4*pi*e*A*B/(I1-I2)

this assumes zero charge in the capacitor to begin with (although could be wrong).

 

[esun]

To elaborate on PM650's response:

I = C dV/dt

I = current
C = capacitance
V = potential difference

The reference potential doesn't matter since it's asking for change in potential, so assume the sphere is initially at zero potential. The capacitance of a conducting sphere is just 4*pi*e*A. The net current into the capacitor is just I1 - I2. So you get:

(I1 - I2) = 4*pi*e*A*dV/dt

B = (I1 - I2)*t/(4*pi*e*A)

t = 4*pi*e*A*B/(I1 - I2)

 

[Farmer]

I think the only hard part of this problem is finding the capacitance of a conducting sphere.

For a symmetric conducting sphere of with a charge q, equally distributed about it's surface, we can treat it as a point charge at the center of the sphere if we're outside the sphere, so the potential at radius A is V(A) = kq/A (same as point charge). Then dV/dt = k/A (dQ/dt) = (k/A) I. In this case, I = (I1-I2). So we have

dV/dt = (k/A) (I1-I2)

Now integrate up once. First the left hand side: (integral time = 0 to time = tfinal) dV/dt = Vfinal = B. Now for the right hand side; since I1, I2, and A are constant in time, this is easy: (integral time = 0 to time = tfinal) (k/A) (I1-I2) = (k/A)(I1-I2)(tfinal). So we get

B = (k/A)(I1-I2)(tfinal)
tfinal = BA/k(I1-I2)

In MKS, k is 1/4(pi)(epsilon), so you get what both PM650 and esun reported.

 

[Estrella]

Originally posted by: esun
To elaborate on PM650's response:

I = C dV/dt

I = current
C = capacitance
V = potential difference

The reference potential doesn't matter since it's asking for change in potential, so assume the sphere is initially at zero potential. The capacitance of a conducting sphere is just 4*pi*e*A. The net current into the capacitor is just I1 - I2. So you get:

(I1 - I2) = 4*pi*e*A*dV/dt

B = (I1 - I2)*t/(4*pi*e*A)

t = 4*pi*e*A*B/(I1 - I2)

Thx for the post! Even though I didn't completely specify what not to say. It would have been sufficient to say that the sphere forms a capacitor(even though there is not another plate.) That seemed to be the part I was missing. I only saw sphere and went oh no! Instead, I should of thought it forms a capacitor with the other plate/shell as r approaches infinity. He briefly said an equivalent statement in class and I only briefly remembered it(obviously).

 

[Farmer]

I hate the "plate at infinity" analogy. Capacitance C should just be the number that relates dV/dt to dQ/dt. That's it, a number which can vary with certain things, it shouldn't have anything to do with plates or spheres or infinity.

 

[esun]

Originally posted by: Farmer
I hate the "plate at infinity" analogy. Capacitance C should just be the number that relates dV/dt to dQ/dt. That's it, a number which can vary with certain things, it shouldn't have anything to do with plates or spheres or infinity.

That's a contradiction of a statement. You want capacitance to relate dV/dt to dQ/dt but have nothing to do with plates or spheres...but the relationship between dV/dt and dQ/dt depends on plates and spheres. Sure, you could always find C experimentally, but you're still finding something related to plates and spheres even if you choose to ignore that.

 

[Farmer]

Originally posted by: esun

Originally posted by: Farmer
I hate the "plate at infinity" analogy. Capacitance C should just be the number that relates dV/dt to dQ/dt. That's it, a number which can vary with certain things, it shouldn't have anything to do with plates or spheres or infinity.

That's a contradiction of a statement. You want capacitance to relate dV/dt to dQ/dt but have nothing to do with plates or spheres...but the relationship between dV/dt and dQ/dt depends on plates and spheres. Sure, you could always find C experimentally, but you're still finding something related to plates and spheres even if you choose to ignore that.

Sorry for not being clearer, I mean it should have nothing to do with plates or spheres specifically, but instead is something that will apply to any arbitrary geometry. To talk about a "plate at infinity" in order to use the concept of capacitance on a spherical conductor is misleading and unnecessary, and only a by product of that fact that parallel-plate capacitors are always used to introduce the concept of capacitance.

The "plate at infinity" just means "I set ground to infinity" or "I define zero potential at infinity." You don't have to say "plate at infinity," which is something that confused me greatly in high school, "what is a plate at infinity?"