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how many time constants (may be a decimal number) must elapse before a capacitor in a series rc circuit is charged to 80.0% of its maximum charge (i.e. you need to find t/rc)?

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How many time constants must elapse before a capacitor is charged?

The equation of charging capacitor is:

[tex]Q=Q_{0} (1-e^{-\frac{t}{r} })[/tex]

Where,

[tex]Q_{0} =[/tex] equilibrium charge

[tex]t=[/tex] time constant

[tex]r=[/tex] time elapse

  • First, we need to set the [tex]Q[/tex] to [tex]Q = 0.80Q_{0}[/tex], so the equation becomes,
  • [tex]Q = 0.80Q_{0}[/tex] = [tex]Q_{0} (1-e^{-\frac{t}{r} } )[/tex]
  • Then, we can eliminate the [tex]Q_{0}[/tex]
  • 0.80 = [tex]1-e^{-\frac{t}{r} }[/tex]
  • Substracting both sides by 1
  • 0.80-1 =  [tex]1-e^{-\frac{t}{r} }[/tex]
  • -0.20 =  [tex]-e^{-\frac{t}{r} }[/tex]
  • [tex]e^{-\frac{t}{r} }[/tex] = 0.20

After that, we need to determine [tex]\frac{t}{r}[/tex] by taking natural log to both sides

  • ㏑ [tex]e^{-\frac{t}{r} }[/tex] = ㏑ 0.20
  • [tex]-\frac{t}{r}[/tex] = ㏑ 0.20
  • [tex]t = -r[/tex]㏑ 0.20
  • [tex]t = 1.609r[/tex]

The number of time constants must elapses are  [tex]\frac{t}{r}=1.609[/tex]

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