RC Circuits

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Discussion

charging

discharging

Start with Kirchhoff's circuit law.

V = IR +  q C

0 = IR +  q C

Turn it into a first order differential equation.

V =  dq  R +  q dt C

0 =  dq  R +  q dt C

Rearrange into a form suitable for the separation of variables procedure.

− RC  dq  = q − CV dt       dq  = −  dt q − CV RC

− RC  dq  = q dt         dq  = −  dt   q RC

Integrate over the appropriate limits.

q       t     ⌠ ⌡ dq    = −  ⌠ ⌡ dt q − CV RC   0       0   ln ⎛ ⎝ q − CV ⎞ ⎠  = −    t − CV RC

q       t     ⌠ ⌡ dq    = −  ⌠ ⌡ dt q RC   Q0       0   ln ⎛ ⎝ q ⎞ ⎠  = −    t Q0 RC

Eliminate the logarithm.

q − CV  = −  e − t / RC     − CV

q  =   e − t / RC     Q0

Solve for charge as a function of time.

q = CV ⎛ ⎝ 1 −  e − t / RC     ⎞ ⎠

q = Q0  e − t / RC

Take the first derivative of charge to get the current.

I =  dq  =  d   ⎡ ⎣ CV ⎛ ⎝ 1 −  e − t / RC     ⎞⎤ ⎠⎦ dt dt

I =  dq  =  d  Q0  e − t / RC     dt dt

As here it is.

I = +  V  e − t / RC     R

I = −  Q0  e − t / RC     RC

Summary

• bullet

Problems

practice

1. Write something.
   • Answer it.
2. Write something.
   • Answer it.
3. Write something.
   • Answer it.
4. Write something completely different.
   • Answer it.

statistical

1. incomplete-discharge.txt
   One fine day, a group of students was conducting an RC circuit experiment. They recorded the potential difference across the capacitor (V₁) and the resistor (V₂) as functions of time as they discharged it. Unfortunately, they did not allow the capacitor to fully discharge. How many time constants passed during the discharge phase of this experiment?

Resources

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