Capacitors: Charge, Voltage, and Current Relationships

Definition: Capacitors store charge, with capacitance relating to:
Permittivity
Surface area of plates
Voltage across the capacitor
Distance between the plates
Capacitance: Defined as the ratio of charge (Q) stored to voltage (V) applied:
Formula: C = Q/V
Units: Farads (F) which represent coulombs per volt

Starting from the capacitance relationship:
Rearranging gives Q = C × V
Current (I): Defined as the rate of change of charge over time:
Formula: I = dQ/dt
Differentiating charge with respect to time:
Yielding I = C × (dV/dt)
Highlights current is proportional to the rate of voltage change, contrasting with resistors where current is directly proportional to voltage.

A capacitor can have voltage across it even when current is zero if the voltage is constant.
Capacitor acts as an open circuit under constant voltage: no current flows despite potential difference.

Instantaneous change in voltage would require infinite current, which is impossible.
Voltage must change over time; rapid voltage changes correspond to rapid current changes but cannot happen instantaneously.
Analogy: like filling a bucket, it takes time to fill from 0 to 10 gallons.

Starting from I = C × (dV/dt), can derive voltage as a function of time:
dV = (1/C) × I dt
Integrating both sides:
Left side: Integrate dV from initial voltage (V0) to voltage at time t (V(t)).
Right side: Integral of current with respect to time from 0 to t.
Resulting expression:
V(t) - V0 = (1/C) × ∫(0 to t) I(x) dx
Final formula for voltage as a function of current:
V(t) = (1/C) ∫(0 to t) I(x) dx + V0
Emphasizes that voltage changes over time as current flows and begins at an initial voltage.

Key Formulas:
Current vs. Voltage in a Capacitor:
- I = C × (dV/dt)
- V(t) = (1/C) ∫(0 to t) I(x) dx + V0
These formulas establish the relationship between current and voltage, highlighting distinct behavior of capacitors compared to resistors.