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Two metallic spheres attached by a long wire. Watch

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    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,
    

q_2=Q_2\left(1-e^{\left(-\alpha t\right)}\right)\ \mathrm{with}\ \alpha=\dfrac{1}{4\pi\varepsilon  _0}\left(\dfrac{R_1+R_2}{R_1R_2} \right)\dfrac{1}{R}

    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)
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    (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,
    

q_2=Q_2\left(1-e^{\left(-\alpha t\right)}\right)\ \mathrm{with}\ \alpha=\dfrac{1}{4\pi\varepsilon  _0}\left(\dfrac{R_1+R_2}{R_1R_2} \right)\dfrac{1}{R}

    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)]?
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    (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
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    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
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    (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. :confused: )
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    (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. :confused: )
    I have it down to,
     \dfrac{\mathrm{d}q_2}{\mathrm{d}  t}=\dfrac{q_2(R_1-R_2)+QR_2}{8\pi\varepsilon_0R_1R  _2R}

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

    EDIT: I said that  q_1+q_2=Q\Rightarrow \dot{q}_1=-\dot{q}_2 to make the equations uncoupled
 
 
 
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