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# Two metallic spheres attached by a long wire. watch

1. Two solid metallic spheres with radii R_1 and R_2 are fixed a large distance
apart (large enough that electric fields due to one have a negligible influence
on the other). A long wire with resistance R is connected between them and after a long time the charge on sphere 1 is Q1 and the charge on sphere 2 is Q2. Show that the charge on sphere 2, q2 as a function of time is given by,

I'm having great difficulty setting up the differential equations, can somebody please help me? (I know this is analogous to discharging and charging capacitors)
2. (Original post by KeyFingot)
Two solid metallic spheres with radii R_1 and R_2 are fixed a large distance
apart (large enough that electric fields due to one have a negligible influence
on the other). A long wire with resistance R is connected between them and after a long time the charge on sphere 1 is Q1 and the charge on sphere 2 is Q2. Show that the charge on sphere 2, q2 as a function of time is given by,

I'm having great difficulty setting up the differential equations, can somebody please help me? (I know this is analogous to discharging and charging capacitors)
A quick question.
I take it you know how to set up the differential equation for the case of a single capacitor discharging through a resistor R, and you know how to get the equation from that of the case of a capacitor charging up? The one with the [1 - e-t/(CR)]?
3. (Original post by Stonebridge)
A quick question.
I take it you know how to set up the differential equation for the case of a single capacitor discharging through a resistor R, and you know how to get the equation from that of the case of a capacitor charging up? The one with the [1 - e-t/(CR)]?
Yes I'm fine with all that
4. I think I'm struggling with this because (If I'm correct) I end up with two variables in my DE and can't seem to find a way to get rid of the other variable.

The two variables being the charges (q1 and q2) as functions of time on the spheres
5. (Original post by KeyFingot)
I think I'm struggling with this because (If I'm correct) I end up with two variables in my DE and can't seem to find a way to get rid of the other variable.

The two variables being the charges (q1 and q2) as functions of time on the spheres
Well
1. the total charge q1 + q2 is constant.
2. The charge Q on a sphere is related to its capacitance via Q=CV, of course. The capacitance of a solid sphere C is 4πεR
3. After a "long time" the discharge/charge stops when the potential of both spheres is equal.
So at the end the spheres are at the same potential where V=Q/C
You know Q1 + Q2 is constant and C is the total capacitance.

I have also noticed that if you look at the formula you have to derive, the value of α is 1/CR in the single capacitor case. ( e-t/CR)
In this case you have two capacitors which are effectively in parallel (they are at the same pd) and the formula for adding two capacitors in parallel is C1 + C2, which in this case is 4πεR1 + 4πεR2
If you sub this formula for the total capacitance into the standard formula, you get the value of α required. (I'm not sure why it works though. )
6. (Original post by Stonebridge)
Well
1. the total charge q1 + q2 is constant.
2. The charge Q on a sphere is related to its capacitance via Q=CV, of course. The capacitance of a solid sphere C is 4πεR
3. After a "long time" the discharge/charge stops when the potential of both spheres is equal.
So at the end the spheres are at the same potential where V=Q/C
You know Q1 + Q2 is constant and C is the total capacitance.

I have also noticed that if you look at the formula you have to derive, the value of α is 1/CR in the single capacitor case. ( e-t/CR)
In this case you have two capacitors which are effectively in parallel (they are at the same pd) and the formula for adding two capacitors in parallel is C1 + C2, which in this case is 4πεR1 + 4πεR2
If you sub this formula for the total capacitance into the standard formula, you get the value of α required. (I'm not sure why it works though. )
I have it down to,

It's pretty horrible looking, is this correct so far?

EDIT: I said that to make the equations uncoupled

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