Electromagnetism · Chapter 14

DC Circuits

The laws governing steady currents in resistive networks — Ohm's law and Kirchhoff's rules — reduce any circuit, no matter how complex, to a system of linear equations.

Prerequisites

Electric fields and potential (Ch. 13) \cdot Basic algebra — no calculus needed for DC circuit

Learning Goals

Distinguish conventional current from electron drift and connect current to charge flow.
Use Ohm's law and power formulas to solve resistor-network problems.
Reduce series and parallel resistor combinations without confusing voltage and current rules.
Apply Kirchhoff's junction and loop rules to circuits that cannot be simplified directly.
Explain how internal resistance changes a real battery's terminal voltage under load.

14.1 Electric Current and Resistance

When a potential difference (voltage) is applied across a conductor, charge carriers — electrons in metals — drift in the direction opposite to the field. The rate of charge flow is the electric current:

I = \frac{dQ}{dt}

(14.1)

Current is measured in amperes (A = C/s). By convention, the direction of current is the direction positive charges would flow — opposite to actual electron motion in metals.

In this chapter every circuit is assumed to have reached a steady state: currents are constant in time, charge does not pile up at junctions, and capacitors or inductors are not changing the current. That is why the algebraic rules below are enough.

Definition 14.1 — Ohm's Law

For many materials over a wide range of conditions, the current through a conductor is proportional to the voltage across it:

V = IR

where R is the resistance

in ohms (\Omega = V/A). Materials that obey this relationship are called ohmic. The resistance

depends on the material (resistivity \rho), length

L

, and cross-sectional area

A

R = \rho L/A

Power dissipated in a resistor (converted to heat) follows from $P = IV$ combined with Ohm's law:

P = IV = I^2R = \frac{V^2}{R}

(14.2)

14.2 Series and Parallel Combinations

Resistors in series carry the same current; their resistances add directly. Resistors in parallel share the same voltage; their reciprocals add.

Theorem 14.1 — Equivalent Resistance

R_\mathrm{eq} = R_1 + R_2 + R_3 + \cdots

\frac{1}{R_\mathrm{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \cdots

For two resistors in parallel:

R_\mathrm{eq} = \frac{R_1R_2}{R_1 + R_2}

Figure 14.1. Interactive circuit simulator. Compare series and parallel modes: in series, every resistor carries the same current and voltage divides; in parallel, every branch has the same voltage and current divides. Adjust the battery voltage and resistances, then check whether the animated charge flow matches those rules.

Example 14.1 — Series-Parallel Network

$R_{1} = 6 \Omega and R_{2} = 3 \Omega are in parallel; this combination is in series with R_{3} = 2 \Omega. A 12 V$ battery is connected. Find the current through each resistor.

Parallel equivalent:

R_{12} = \frac{6 \times 3}{6 + 3} = \frac{18}{9} = 2\,\Omega

Total resistance:

R_\mathrm{total} = R_{12} + R_3 = 2 + 2 = 4\,\Omega

Total current:

I = V/R = 12/4 = 3\,\mathrm{A}

(this flows through

R_3

)

Voltage across parallel:

V_{12} = I R_{12} = 3 \times 2 = 6\,\mathrm{V}

Branch currents:

I_1 = V_{12}/R_1 = 6/6 = 1\,\mathrm{A}

I_2 = V_{12}/R_2 = 6/3 = 2\,\mathrm{A}

Check:

I_1 + I_2 = 1 + 2 = 3\,\mathrm{A}

= total ✓

14.3 Kirchhoff's Rules

For circuits too complex to reduce by series/parallel rules, Kirchhoff's two laws provide a systematic approach. They follow directly from charge conservation and energy conservation.

The method is procedural: assign current directions, write one current-conservation equation at a junction, write voltage-conservation equations around independent loops, then solve the resulting linear system. A negative current is not an error; it means the real direction is opposite to the one you guessed.

Definition 14.2 — Kirchhoff's Junction Rule (KCL)

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

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

This is conservation of charge — no charge accumulates at a junction.

Definition 14.3 — Kirchhoff's Loop Rule (KVL)

The sum of all potential changes around any closed loop in a circuit is zero:

\sum \Delta V = 0

(around any closed loop)This is conservation of energy — a charge returning to its starting point gains and loses equal energy. Traversing a resistor in the direction of current:

\Delta V = -IR

Traversing a battery from - to +

\Delta V = +\varepsilon

Example 14.2 — Two-Loop Circuit by Kirchhoff's Rules

$Two batteries (\varepsilon_{1} = 12 V, \varepsilon_{2} = 6 V) and three resistors (R_{1} = 2 \Omega, R_{2} = 4 \Omega, R_{3} = 3 \Omega). \varepsilon_{1}$ $and R_{1} in the left branch, \varepsilon_{2} and R_{2} in the right, R_{3} in the middle. Find I_{1}, I_{2}, I_{3}$

Assign currents:

I_1

(left, upward),

I_2

(right, upward),

I_3

(middle, upward)

Junction rule (top):

I_1 + I_2 = I_3

→

I_3 = I_1 + I_2

Left loop (KVL):

+\varepsilon_1 - I_1R_1 - I_3R_3 = 0

→

12 - 2I_1 - 3(I_1+I_2) = 0

→

5I_1 + 3I_2 = 12

Right loop (KVL):

+\varepsilon_2 - I_2R_2 - I_3R_3 = 0

→

6 - 4I_2 - 3(I_1+I_2) = 0

→

3I_1 + 7I_2 = 6

Solve system:From eq. 1:

I_1 = (12-3I_2)/5

. Substitute:

3(12-3I_2)/5 + 7I_2 = 6

→

I_2 = -0.47\,\mathrm{A}

Result:

I_1 \approx 2.68\,\mathrm{A}

I_2 \approx -0.47\,\mathrm{A}

(flows opposite to assumed direction),

I_3 \approx 2.21\,\mathrm{A}

14.4 EMF and Internal Resistance

A real battery is not a pure voltage source — it has internal resistance r. The terminal voltage $V_t$ differs from the EMF $\varepsilon$ whenever current flows:

V_t = \varepsilon - Ir \qquad \text{(discharging)}

(14.3)

This means the terminal voltage drops under load. A car battery rated at 12 V might deliver only 10 V while cranking the engine (drawing 200 A through r ≈ 0.01 Ω). To maximize power transfer to an external load $R_L$ , set $R_L = r$ (maximum power transfer theorem).

Definition 14.4 — Common Traps

Current is not used up: charge flow is conserved at junctions; energy is dissipated, not charge.
Series and parallel rules swap what stays the same: series means same current; parallel means same voltage.
Power formulas require local values: $in P = V^{2}/R, V must be the voltage across that resistor, not automatically the battery vol$ age.
Negative Kirchhoff currents are useful: they reveal an incorrect guessed direction, not a failed solution.
Real batteries sag under load: terminal voltage is lower than EMF whenever current flows out through internal resistance.

Exercises — 14.1–14.4 DC Circuits

A 12 V battery is connected to R_{1} = 3 \Omega and R_{2} = 1 \Omega in series. Find the current

Straightforward

Find the equivalent resistance of 4 \Omega and 6 \Omega connected in parallel

Straightforward

A 9 V battery drives R_{1} = 3 \Omega in series with R_{2} = 6 \Omega \parallel R_{3} = 6 \Omega. Find the power dissipate

d in R_{2}

Intermediate

Apply Kirchhoff's rules: \varepsilon_{1} = 10 V, R_{1} = 2 \Omega (left branch); \varepsilon_{2} = 8 V, R_{2} = 3 \Omega (right bran

ch); R_{3} = 5 \Omega (connecting middle). Find all three branch currents

Intermediate

Derive the balance condition for a Wheatstone bridge (R_{1}, R_{2} in one branch; R_{3}, R_{4} in the

other; galvanometer between midpoints). What unknown R_{4} balances the bridge if R_{1} = 100 \Omega

R_{2} = 50 \Omega, R_{3} = 200 \Omega

Challenging

Key Takeaways

$Current I = dQ/dt; Ohm's law: V = IR; power P = I^{2}R = V^{2}/R$ .
Series resistors add; parallel resistors add as reciprocals.
Kirchhoff's junction rule (KCL): charge is conserved at every node.
Kirchhoff's loop rule (KVL): energy is conserved around every loop.
$Real batteries have internal resistance r; terminal voltage V = \varepsilon - Ir drops under load$ .
Maximum power transfer occurs when load resistance equals source internal resistance.