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Topic 7 of 15

DC Circuits

A circuit is a closed path through which current flows continuously, driven by an EMF source such as a battery. Kirchhoff's two laws — charge conservation at every junction, and energy conservation around every loop — give us a complete toolkit to analyze any DC network, no matter how complex.

Key Concepts

EMF and Internal Resistance

An EMF source (battery, generator) maintains a potential difference. A real battery has internal resistance

r

, so its terminal voltage is

V_{\text{term}} = \varepsilon - Ir

, where

\varepsilon

is the EMF and

I

is the current drawn. Terminal voltage drops under load.

Kirchhoff's Current Law (KCL)

At any junction in a circuit, the sum of currents entering equals the sum leaving:

\sum I_{\text{in}} = \sum I_{\text{out}}

. This is conservation of charge — charge cannot accumulate at a node.

Kirchhoff's Voltage Law (KVL)

The sum of all potential differences around any closed loop is zero:

\sum \Delta V = 0

. This is conservation of energy — a charge returning to its starting point gains and loses the same total energy.

Resistors in Series and Parallel

Series: same current through each;

R_{\text{eq}} = R_1 + R_2 + \cdots

. Parallel: same voltage across each;

1/R_{\text{eq}} = 1/R_1 + 1/R_2 + \cdots

. The equivalent parallel resistance is always less than the smallest individual resistor.

RC Circuits

A capacitor charging through a resistor: voltage builds exponentially with time constant

\tau = RC

. At

t = \tau

the capacitor reaches

\approx 63\%

of full charge. After

5\tau

it is essentially fully charged (

>99\%

). Discharging follows the same exponential decay.

Key Equations

Terminal voltage

V_{\text{term}} = \varepsilon - Ir

Actual voltage delivered by a battery with EMF ε and internal resistance r when supplying current I.

Series resistors

R_{\text{eq}} = R_1 + R_2 + \cdots

Equivalent resistance of resistors in series. Same current flows through all.

Parallel resistors

\frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots

Equivalent resistance of resistors in parallel. Same voltage across all.

RC time constant

\tau = RC

Time constant for charging/discharging. After time τ, capacitor is at 63% of final voltage; after 5τ it is >99% charged.

Capacitor charging voltage

V_C(t) = \varepsilon\bigl(1 - e^{-t/\tau}\bigr)

Voltage across capacitor as a function of time when charging from 0 through resistor R from EMF ε.

Worked Example

Series-Parallel Circuit with Internal Resistance

Problem

A $12$ V battery with internal resistance $r = 1\,\Omega$ is connected to $R_1 = 5\,\Omega$ in series with a parallel combination of $R_2 = 6\,\Omega$ and $R_3 = 3\,\Omega$ . Find the total current drawn and the terminal voltage.

Solution

First reduce the parallel combination:

R_{23} = \frac{R_2 R_3}{R_2 + R_3} = \frac{(6)(3)}{6+3} = \frac{18}{9} = 2\,\Omega

Total external resistance and full circuit resistance:

R_{\text{ext}} = R_1 + R_{23} = 5 + 2 = 7\,\Omega \qquad R_{\text{total}} = r + R_{\text{ext}} = 1 + 7 = 8\,\Omega

Total current from the battery:

I = \frac{\varepsilon}{R_{\text{total}}} = \frac{12}{8} = 1.5\text{ A}

Terminal voltage (voltage at battery terminals):

V_{\text{term}} = \varepsilon - Ir = 12 - (1.5)(1) = 10.5\text{ V}

Answer I = 1.5 A; terminal voltage = 10.5 V.

Practice

Exercises

7 problems

1 of 7

Three resistors $R_1 = 4\,\Omega$ , $R_2 = 6\,\Omega$ , and $R_3 = 10\,\Omega$ are connected in series to a $V = 20$ V battery. What current (in A) flows through the circuit?

Your answer

2 of 7

Resistors $R_1 = 6\,\Omega$ and $R_2 = 12\,\Omega$ are connected in **parallel**. What is the equivalent resistance (in Ω)?

Your answer

3 of 7

A battery has EMF $\varepsilon = 9.0$ V and internal resistance $r = 0.50\,\Omega$ . When it delivers current $I = 2.0$ A, what is the terminal voltage (in V)?

Your answer

4 of 7

Resistors $R_1 = 8\,\Omega$ and $R_2 = 4\,\Omega$ are in series with a $12$ V battery. What is the voltage (in V) across $R_2$ ?

Your answer

5 of 7

A current of $I = 3.0$ A flows through a resistor $R = 8.0\,\Omega$ . What power (in W) is dissipated in it?

Your answer

6 of 7

An RC circuit has $R = 500\,\Omega$ and $C = 100\,\mu$ F. What is the time constant $\tau$ (in ms)?

Your answer

7 of 7

A $2\,\Omega$ resistor is in series with a parallel combination of two $4\,\Omega$ resistors, all connected to a $12$ V battery. What is the total current (in A) drawn from the battery?

Your answer

Key Takeaways

KCL (charge conservation) at every node; KVL (energy conservation) around every loop — these two rules solve any DC circuit.
Series resistors share the same current; parallel resistors share the same voltage. Use this to spot which applies.
Terminal voltage drops under load: $V_{\text{term}} = \varepsilon - Ir$ . A battery with high internal resistance is less useful under heavy load.
RC time constant $\tau = RC$ : larger $R$ or larger $C$ means slower charging/discharging.
The voltage divider rule for two series resistors: $V_{R_2} = V_{\text{source}} \cdot R_2/(R_1 + R_2)$ .