AP Physics C: Electricity and Magnetism

Direct Current Circuits

RC Circuits

Learning Objectives

By the end of this lesson, students should be able to:

• Define an RC circuit.

• Explain the charging and discharging behavior of a capacitor.

• Derive the characteristic equations of RC circuits.

• Understand the concept of the time constant.

• Analyze current, charge, and voltage as functions of time.

• Solve AP Physics C problems involving RC circuits.

Introduction to RC Circuits

What is an RC Circuit?

An RC circuit is a circuit containing:

• A resistor ((R))

• A capacitor ((C))

connected in series or in a more complex arrangement.

RC circuits are important because they introduce time-dependent behavior into electric circuits.

Unlike simple resistor circuits, the current in an RC circuit changes with time.

Applications of RC Circuits

RC circuits are used in:

• Camera flashes

• Timing circuits

• Electronic filters

• Signal processing systems

• Computer memory circuits

Understanding RC circuits is an important AP Physics C topic because they combine concepts from:

• Capacitance

• Current

• Energy conservation

• Differential equations

Components of an RC Circuit

Resistor

The resistor opposes the flow of current.

According to Ohm’s Law:

$$
V_R = IR
$$

where:

• (V_R) = voltage across the resistor

• (I) = current

• (R) = resistance

Capacitor

A capacitor stores charge and electrical energy.

The charge stored on a capacitor is:

$$
Q = CV
$$

where:

• (Q) = charge

• (C) = capacitance

• (V) = voltage across the capacitor

Charging an RC Circuit

Circuit Description

Consider:

• A battery of emf (\mathcal{E})

• A resistor (R)

• A capacitor (C)

Initially:

$$
Q=0
$$

The capacitor is uncharged.

When the switch is closed:

• Current begins flowing.

• Charge accumulates on the capacitor.

• Current gradually decreases.

• Eventually the capacitor becomes fully charged.

Kirchhoff’s Loop Rule

Applying Energy Conservation

For the charging circuit:

$$
\mathcal{E}-IR-\frac{Q}{C}=0
$$

where:

$$
V_C=\frac{Q}{C}
$$

is the voltage across the capacitor.

Differential Equation

Since:

$$
I=\frac{dQ}{dt}
$$

we obtain:

$$
\mathcal{E}=R\frac{dQ}{dt}=\frac{Q}{C}0
$$

Solving this differential equation produces the charging equations.

Charge During Charging

Charging Equation

The charge on the capacitor as a function of time is:

$$
Q(t)=C\mathcal{E}\left(1-e^{-t/RC}\right)
$$

Long-Time Behavior

As:

$$
t\rightarrow\infty
$$

the exponential term approaches zero.

Therefore:

$$
Q_{max}=C\mathcal{E}
$$

The capacitor becomes fully charged.

Current During Charging

Current Equation

The current in the circuit is:

$$
I(t)=\frac{\mathcal{E}}{R}e^{-t/RC}
$$

Interpretation

Initially:

$$
t=0
$$

so:

$$
I_0=\frac{\mathcal{E}}{R}
$$

which is the maximum current.

As time increases:

$$
I\rightarrow0
$$

The current gradually disappears because the capacitor opposes further charge flow.

Voltage During Charging

Capacitor Voltage

The voltage across the capacitor is:

$$
V_C(t)=\mathcal{E}\left(1-e^{-t/RC}\right)
$$

Initial and Final Values

Initially:

$$
V_C=0
$$

The capacitor behaves like a wire.

Eventually:

$$
V_C=\mathcal{E}
$$

The capacitor behaves like an open circuit.

The Time Constant

Definition

The quantity:

$$
\tau=RC
$$

is called the time constant.

Physical Meaning

The time constant determines how quickly the capacitor charges or discharges.

After one time constant:

$$
t=\tau
$$

the capacitor reaches approximately:

$$
63%
$$

of its final charge.

Important Values

At:

$$
t=\tau
$$

$$
Q=0.632Q_{max}
$$

At:

$$
t=2\tau
$$

$$
Q=0.865Q_{max}
$$

At:

$$
t=3\tau
$$

$$
Q=0.950Q_{max}
$$

At:

$$
t=5\tau
$$

the capacitor is considered essentially fully charged.

Discharging an RC Circuit

Initial Conditions

Suppose a charged capacitor is disconnected from the battery and connected across a resistor.

Initially:

$$
Q=Q_0
$$

Current begins flowing through the resistor.

Discharge Equation

Applying Kirchhoff’s Loop Rule:

$$
IR+\frac{Q}{C}=0
$$

Substituting:

$$
I=\frac{dQ}{dt}
$$

leads to the discharge equations.

Charge During Discharge

Charge Equation

The charge decreases exponentially:

$$
Q(t)=Q_0e^{-t/RC}
$$

Behavior

As time increases:

$$
Q\rightarrow0
$$

The capacitor gradually loses all stored charge.

Current During Discharge

Current Equation

The magnitude of the discharge current is:

$$
I(t)=\frac{Q_0}{RC}e^{-t/RC}
$$

Interpretation

The current is largest immediately after the discharge begins.

It then decreases exponentially toward zero.

Voltage During Discharge

Capacitor Voltage

The voltage across the capacitor is:

$$
V_C(t)=V_0e^{-t/RC}
$$

where:

$$
V_0=\frac{Q_0}{C}
$$

Long-Time Behavior

Eventually:

$$
V_C\rightarrow0
$$

The capacitor becomes completely discharged.

Graphs of RC Behavior

Charging Graphs

Charge:

$$
Q(t)=Q_{max}\left(1-e^{-t/RC}\right)
$$

starts at zero and approaches a maximum.

Voltage:

$$
V_C(t)=\mathcal{E}\left(1-e^{-t/RC}\right)
$$

has the same shape.

Current:

$$
I(t)=\frac{\mathcal{E}}{R}e^{-t/RC}
$$

starts large and decreases toward zero.

Discharging Graphs

Charge, current, and voltage all follow exponential decay:

$$
e^{-t/RC}
$$

and approach zero as time increases.

Example 1

Finding the Time Constant

A circuit contains:

$$
R=2.0\times10^3\Omega
$$

and

$$
C=5.0\times10^{-6}F
$$

Find the time constant.

Solution

Use:

$$
\tau=RC
$$

Substitute:

$$
\tau=(2.0\times10^3)(5.0\times10^{-6})
$$

$$
\tau=0.010s
$$

Answer

$$
\tau=1.0\times10^{-2}s
$$

Example 2

Charge During Charging

A capacitor is charging in an RC circuit.

The maximum charge is:

$$
Q_{max}=20\mu C
$$

Find the charge after one time constant.

Solution

At:

$$
t=\tau
$$

$$
Q=0.632Q_{max}
$$

Substitute:

$$
Q=(0.632)(20)
$$

$$
Q=12.6\mu C
$$

Answer

$$
Q=12.6\mu C
$$

Common AP Exam Mistakes

Mistake 1

Assuming current remains constant.

In RC circuits:

$$
I
$$

changes continuously with time.

Mistake 2

Confusing charging and discharging equations.

Charging:

$$
1-e^{-t/RC}
$$

Discharging:

$$
e^{-t/RC}
$$

Always identify which process is occurring.

Mistake 3

Forgetting the time constant.

The key parameter controlling RC behavior is:

$$
\tau=RC
$$

AP Free-Response Strategy

Begin with Kirchhoff’s Loop Rule

Most RC derivations start from:

$$
\sum\Delta V=0
$$

This often earns immediate partial credit.

Memorize the Three Charging Equations

Charge:

$$
Q=C\mathcal{E}
\left(
1-e^{-t/RC}
\right)
$$

Voltage:

$$
V_C=\mathcal{E}
\left(
1-e^{-t/RC}
\right)
$$

Current:

$$
I=
\frac{\mathcal{E}}{R}
e^{-t/RC}
$$

These equations appear frequently on AP exams.

Summary

Key Takeaways

• RC circuits contain both a resistor and a capacitor.

• The time constant is:

$$
\tau=RC
$$

• During charging:

$$
Q=C\mathcal{E}
\left(
1-e^{-t/RC}
\right)
$$

$$
I=
\frac{\mathcal{E}}{R}
e^{-t/RC}
$$

$$
V_C=
\mathcal{E}
\left(
1-e^{-t/RC}
\right)
$$

• During discharging:

$$
Q=Q_0e^{-t/RC}
$$

$$
I=
\frac{Q_0}{RC}
e^{-t/RC}
$$

$$
V_C=V_0e^{-t/RC}
$$

• RC circuits exhibit exponential behavior.

• The time constant determines the rate of charging and discharging.

• RC circuits are one of the most important applications of differential equations in AP Physics C: Electricity and Magnetism.