What is the Charge on the Capacitor in a Circuit After Switching Positions?

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SUMMARY

The charge on the capacitor in the circuit after switching from position "a" to "b" at t=0s is calculated using the equations C=Q/V, V=IR, and q=Qinit.(1-e^(-t/τ)). The initial charge Q is determined to be 36μC, and the current I is 0.36A. However, the correct charge at t=50s is 22μC, which can be derived by substituting the time constant τ=RC into the charge equation. This highlights the importance of understanding the time-dependent behavior of capacitors in RC circuits.

PREREQUISITES
  • Understanding of capacitor charging equations
  • Familiarity with Ohm's Law (V=IR)
  • Knowledge of time constant in RC circuits (τ=RC)
  • Basic calculus for exponential functions
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  • Study the derivation of the capacitor charging equation q=Qinit.(1-e^(-t/τ))
  • Learn about the implications of the time constant τ in RC circuits
  • Explore the behavior of capacitors in transient analysis
  • Investigate practical applications of capacitors in electronic circuits
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Homework Statement


The switch in the figure has been in position a for a long time. It is changed to position "b" at t=0s. What is the charge Q on the capacitor at t=50s?

(zeros denote empty space in the circuit)

|----(a)0000(b)--------- |
|00000\00000000000000|
9V00000\000000000000|
|00000000|00000000025 Ohm
|00000004uF000000000|
|________|___________|


Homework Equations


(1) C=Q/V
(2) V=IR
(3) q=Qinit.(1-e^(-t/τ))
(4) τ=RC


The Attempt at a Solution


Ok, so I start by using (1) to find that Q=36uC, then I use (2) to find that I=.36A, but after that is where i encounter problems. I am trying to use (3) to find the charge but it's not giving me the right answer. I'm not sure how to use (4), what is that formula telling me, and am i doing everything right?

The answer should be 22uC
 
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Notice that in equation (3) you have the parameter 1/T, this is exactly what equation (4) is telling you. Try substituting (4) into (3) and seeing if you get the right solution.
 
You may want to take a closer look at your equation (3). In particular, check what the function yields for t = 0 and in the limit as t --> infinity. Does it correspond to what you intuitively feel should happen to the charge on the capacitor?
 

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