Capacitor charging through a resistor

When an initially uncharged capacitor is connected to a constant-voltage supply through a resistor, charge builds up on the capacitor. Its potential difference rises from zero towards the supply voltage.

The supply voltage is shared between the capacitor and the resistor. As the capacitor’s potential difference rises, the potential difference across the resistor falls. The charging current therefore decreases. It approaches zero as the capacitor’s potential difference approaches the supply voltage.

Charge and voltage during charging

For resistance $$R$$ and capacitance $$C$$, the charge $$Q$$ at time $$t$$ is:

Here, $$Q_\infty$$ is the final charge approached by the capacitor. The capacitor potential difference $$V$$ follows the same pattern:

Here, $$V_\infty$$ is the final capacitor potential difference, equal to the supply voltage, and $$Q_\infty = CV_\infty$$. Both $$Q$$ and $$V$$ start at zero, rise steeply at first, then increase more slowly. Their graphs approach their final values asymptotically: they get closer and closer without reaching them in the ideal mathematical model.

Charging current

Initially, the uncharged capacitor has zero potential difference across it, so the full supply voltage is across the resistor. The current is then at its maximum, $$I_0 = V_\infty/R$$. At time $$t$$, the current is:

The current graph starts at $$I_0$$ and falls steeply at first, then approaches zero asymptotically. The product $$RC$$ is the time constant: a larger value means that the charge and voltage rise, and the current falls, more slowly.

Capacitor discharge through a resistor

When a charged capacitor is connected across a resistor, charge flows through the resistor. The capacitor loses charge, so its potential difference falls. The smaller potential difference drives a smaller current through the resistor, so the capacitor loses charge more slowly as the discharge continues.

For a fixed capacitance $$C$$, the capacitor’s charge and potential difference are proportional: $$Q = CV$$. With a fixed resistance $$R$$, the current magnitude is proportional to the potential difference across the resistor. Charge, voltage and current magnitude therefore all decrease by the same fraction in a given time interval. This is exponential decay.

Discharge equations

If $$Q_0$$ and $$V_0$$ are the initial charge and voltage, and $$I_0 = V_0/R$$ is the initial current magnitude, their values at time $$t$$ are:

Here $$I$$ is the magnitude of the discharge current. The product $$RC$$ is the circuit’s time constant: after a time $$RC$$, each quantity has fallen to $$1/\mathrm{e}$$, or about 37%, of its initial value. A larger $$R$$ or $$C$$ gives a longer time constant and a slower discharge.

Discharge graphs and current direction

Graphs of charge, voltage and current magnitude against time fall steeply at first, when the current is largest. They then become less steep and approach zero asymptotically: the exponential model approaches zero without reaching it at a finite time.

Discharge current flows in the opposite direction to charging current. If charging current is taken as positive, a graph of signed discharge current lies below the time axis and approaches zero from below. Its magnitude still follows the positive current equation above.

Interpreting capacitor charge, voltage and current graphs

A capacitor’s graphs can show how its charge changes with time. The gradient of a charge–time or voltage–time graph tells us about the current at a particular instant. The signed area under a current–time graph tells us how much the charge changes over an interval.

Gradients of charge–time and voltage–time graphs

The gradient of a charge–time graph is the rate of change of charge. It gives the signed current:

To find the current at an instant, find the gradient of the tangent to the graph at that point. A positive gradient gives a positive current; a negative gradient gives a negative current. A steeper gradient means a greater current magnitude, regardless of its sign. A horizontal tangent gives zero current at that instant.

For a capacitor with constant capacitance, $$Q=CV$$. Its charge and voltage therefore change in proportion, so the voltage–time gradient also tells us about the current:

Thus, for a given capacitor, a steeper voltage–time graph also indicates a greater current magnitude. The signs assume that the chosen current direction and voltage polarity are consistent with the charge being measured.

Area under a current–time graph

The signed area between a current–time graph and the time axis gives the change in charge over the interval:

Area above the time axis gives a positive change in charge; area below it gives a negative change under the chosen current sign convention. If the graph has areas on both sides of the axis, combine them with their signs to find the net change.

For example, a constant current of $$-2,\mathrm{A}$$ for $$3,\mathrm{s}$$ gives a signed area of $$-6,\mathrm{C}$$. The charge therefore changes by $$-6,\mathrm{C}$$.

Checking graph axes and units

The meaning of a gradient or area depends on the quantities on the axes. A charge–time gradient has units of $$\mathrm{C,s^{-1}}$$, equivalent to amperes. A current–time area has units of $$\mathrm{A,s}$$, equivalent to coulombs.

An area under a voltage–time graph has units of $$\mathrm{V,s}$$, while an area under a charge–time graph has units of $$\mathrm{C,s}$$. Neither directly gives the change in capacitor charge.

RC time constant and half-time

In a resistor–capacitor circuit, the time constant sets the timescale over which charge, voltage and current change. A larger time constant means a slower change. It is the product of the resistance $$R$$ and the capacitance $$C$$:

The time constant $$\tau$$ is measured in seconds. Increasing either $$R$$ or $$C$$ increases $$\tau$$, so charging or discharging takes longer.

The time constant on a graph

The changes are exponential: the quantity changes rapidly at first, then more slowly as it approaches its final value.

• Decay: After one time constant, a decaying charge, voltage or current magnitude has fallen to $$\mathrm{e}^{-1}$$, approximately 37% of its initial value.
• Charging: After one time constant, the charge and capacitor voltage have risen to approximately 63% of their final values, starting from zero.

These fractions allow $$RC$$ to be found from a time graph. For example, on a decay graph, find 37% of the initial value and read the time taken to reach it. That time is one time constant.

Alternatively, draw a tangent to the curve at the start and extend it. For a decay curve, the tangent meets the zero level after one time constant. For a rising charging curve, the tangent meets the final-value level after one time constant.

Half-time

The half-time is the time taken for an exponentially decaying quantity to fall to half its value. For an RC circuit:

The half-time is constant for a given circuit: each successive half-time reduces the remaining decaying quantity by half. During charging, the same interval halves the remaining difference between the charge or capacitor voltage and its final value.