RC Circuits
Discussion
Start with Kirchhoff's circuit law.
Turn it into a first order differential equation.
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.
Solve for charge as a function of time.
Take the first derivative of charge to get the current.
| I = |
dq |
= |
d |
|
⎡
⎣ |
CV |
⎛
⎝ |
1 − |
|
⎞⎤
⎠⎦ |
| dt |
dt |
|
|
As here it is.
