A capacitor stores electrical energy in an electric field; its capacitance C relates charge Q to voltage: Q = CV. The current through a capacitor is proportional to the rate of change of voltage: i = C(dv/dt). Capacitors act as open circuits to DC steady state but pass AC signals; they oppose sudden voltage changes.
Build simple RC circuits and observe charging with a meter or oscilloscope. Derive the differential equation for charging from first principles using Kirchhoff's voltage law and the definition of capacitive current.
From your study of circuit element types, you know that resistors dissipate energy according to Ohm's law (V = IR), while capacitors belong to a different category — they store energy rather than dissipate it. The physical picture is two conducting plates separated by an insulating gap. When you apply a voltage, charge accumulates on the plates: positive charge on one side, negative on the other. Capacitance C is the proportionality between the stored charge Q and the voltage V across the plates: Q = CV. A larger capacitance means more charge can be stored per volt — physically, larger plates or thinner insulation.
The current-voltage relationship for a capacitor follows directly from this definition. Current is the rate of flow of charge: i = dQ/dt. Substituting Q = CV gives i = C(dv/dt). Read this carefully: current through a capacitor is proportional to the *rate of change* of voltage, not to voltage itself. This one equation explains nearly everything about capacitor behavior. If voltage is constant (DC steady state), dv/dt = 0, so current is zero — the capacitor looks like an open circuit. If voltage is changing rapidly (high-frequency AC), dv/dt is large, so current is large — the capacitor passes current easily. This is why capacitors block DC and pass AC.
The equation i = C(dv/dt) also reveals that a capacitor opposes sudden changes in voltage. To change the voltage by a finite amount instantaneously would require an infinite current — physically impossible in real circuits with non-zero source resistance. This makes capacitors natural "voltage memory" elements: the voltage across a capacitor at any moment is the integral of all the current that has flowed through it up to that point. Energy is stored in the electric field between the plates: E = ½CV². Unlike a resistor, which converts energy to heat, a capacitor holds that energy and can return it to the circuit later.
In circuit analysis, this behavior means capacitors introduce dynamics — differential equations rather than algebraic ones. The simple RC circuit you will study next is the prototype: a resistor limits how fast current flows, and the capacitor integrates that current into a slowly-changing voltage. The resulting exponential charging and discharging curves (with time constant τ = RC) are the fundamental transient response of first-order circuits, and they appear everywhere from filter design to timing circuits to the modeling of biological membranes.