How Does Capacitor Separation Affect Its Half-Life in a Circuit?

AI Thread Summary
The discussion focuses on calculating the half-life of exponential decay in a circuit with a capacitor, specifically considering the effect of capacitor separation. The relevant formula for exponential decay is Q = Qo*e^(-t/RC), where Q is the charge at time t, and Qo is the initial charge. To find the half-life, the value of Q is set to Qo/2, and logarithmic manipulation is applied. The capacitance is expressed as C = ε₀*A/d, linking the separation distance (d) to the half-life calculation. Ultimately, the discussion aims to derive an expression for T1/2 that incorporates the separation of the capacitor plates.
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Homework Statement



If I have a circuit in which the capacitor is the large disk device and for each value of separation=d, calculate the half-life of exponential decay?

Homework Equations





The Attempt at a Solution



What is the formula to do this?
 
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Formula for exponential decay is Q = Qo*e^-t/RC
At half life Q = Qo/2.Substitute this value in the above expression. Take the logarithm and substitute C = epsilon(not)*A/d.Then write down the expression for T1/2.
 
Thread 'Variable mass system : water sprayed into a moving container'

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Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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