Helpful Circuit Stuff

Circuits with resistors:

In series:

R_eq=R₁+R₂+R₃+R₄+...

I=I₁=I₂=I₃=I₄=...

V=IR₁+IR₂+IR₃+IR₄+...

In parallel:

1/R_eq=1/R₁+1/R₂+1/R₃+1/R₄+...

I=I₁+I₂+I₃+I₄...

V=V₁=V₂=V₃=V₄=...

Note: Replacing resistors in parallel with one resistor of equivalent total resistance is very useful when analyzing circuits

Circuits with capacitors:

In series:

1/C_eq=1/C₁+1/C₂+1/C₃+1/C₄+...

Q=Q₁=Q₂=Q₃=Q₄=... (Conservation of charge)

V=V₁+V₂+V₃+V₄+...

In parallel:

C_eq=C₁+C₂+C₃+C₄+...

Q=Q₁+Q₂+Q₃+Q₄... (Conservation of energy)

V=V₁=V₂=V₃=V₄=...

After a long time, a capacitor acts like an open spot in the circuit; current through the section of the circuit with the capacitor= 0 A

RC Circuits:

Time constant=RC, or Greek letter tau

I=-dQ/dt

When a resistor and capacitor are in parallel, voltage drop across resistor=voltage drop across capacitor.

When a resistor and capacitor are in series, current is the same through the resistor as through the capacitor.

Charging RC Circuit: Current decreases over time. Charge on capacitor and potential drop across capacitor increase over time.

*In the long run,V_C=V_T (V-terminal=V-capacitor)

*I=(V_T/R)e^-t/RC

*Q=Q_f(1-e^-t/RC)

*Q_f=CV_T

Discharging a capacitor: Current flows from the positive plate of capacitor to the negative plate and through the resistor. Current, charge, and voltage decrease over time.

*I=I₀e^-t/RC

*Q=Q₀e^-t/RC

*I₀=V₀/R=Q₀/(RC)