Is a capacitor really linear?

Question

In basic electronics, we’re taught that capacitors are linear components. But when analyzing the charging or discharging behavior of a capacitor (like in an RC circuit), the voltage vs. time graph is clearly exponential — which looks nonlinear. This seems confusing: if the response is exponential, doesn’t that imply nonlinearity? Why is a capacitor still classified as a linear element despite this?

I’d appreciate some clarity on how this aligns with the definition of linearity in circuit theory.

Vikas · Answer

In circuit theory, a capacitor is considered linear because the relationship between its current and voltage is defined by a linear differential equation:

This equation is linear because:

The current is directly proportional to the rate of change of voltage.
It satisfies the principles of superposition and homogeneity (doubling input → doubles output).

The exponential voltage-time response seen in RC circuits is the solution of a linear system — not a sign of nonlinearity. So, even though the time-domain response isn't a straight line, the capacitor's behavior remains linear.

Admin · Answer

A capacitor is called linear because the relationship between the voltage across it and the current flowing through it is linear. The exponential curve you're seeing is its behavior over time, which is different.

Here’s the breakdown:

What "Linear" actually means here

In circuit theory, a component is linear if it follows the rule of superposition and scaling.¹ In simple terms: if you double the cause, you double the effect.

For a capacitor, the relationship is defined by the equation ² $I = C \frac{d V}{d t}$ .³ This means the current (⁴ $I$ ) is directly proportional to the rate of change of voltage (⁵ $d V / d t$ ).⁶

So, if you double the current going into the cap, its voltage changes twice as fast.
If you halve the current, its voltage changes half as fast.

That direct, proportional relationship is what makes it a linear component.

So why the exponential curve?

That famous exponential curve shows the capacitor's voltage versus time when it's part of a circuit with a resistor (an RC circuit). It's not a direct graph of voltage vs. current.

Think about what happens when you charge it:

At the start, the capacitor is empty, so a large current flows in.
As it charges, voltage builds up across it.
This built-up voltage opposes the source, which reduces the voltage across the resistor, and therefore reduces the current flowing into the cap.
So, the charging slows down as it gets fuller.

This process of "charging slower and slower as it fills up" is what creates that exponential curve. The capacitor itself is still behaving linearly at any given instant, but the behavior of the whole circuit over time is exponential.

So:

Component's V-I relationship: Linear. (The physics of the cap itself).
Circuit's V-T response: Exponential. (The behavior you see over time in an RC circuit).

Hope that clears it up!