DC Circuits

Concise answer

Reduce resistor networks with the series and parallel rules, fall back on Kirchhoff's junction and loop rules when the network will not reduce, and treat every RC transient as an exponential with time constant $\tau = RC$ .

Definitions

Ohm's law: For an ohmic device, the relation $V = IR$ between the voltage across the device, the current through it, and its resistance.
Equivalent resistance: The single resistance that could replace a network of resistors while drawing the same current from the source.
Junction rule: Kirchhoff's statement that the total current entering a junction equals the total current leaving it (charge conservation).
Loop rule: Kirchhoff's statement that the potential changes around any closed loop sum to zero (energy conservation).
Time constant: The characteristic time $\tau = RC$ over which an RC circuit's charge, current, or voltage changes by a factor of $e$.

Intuition

Series elements share one current, so their voltage drops and resistances add. Parallel elements share one voltage, so their currents add — which is why adding a parallel branch always lowers the equivalent resistance.

Kirchhoff's rules are just conservation laws in circuit clothing: charge does not pile up at junctions, and a charge carried around a closed loop returns with no net energy change.

Concept walkthrough

Ohm's law, $V = IR$ , holds for ohmic materials in which resistance is independent of the applied voltage. Combined with the power relations $P = IV = I^2R = V^2/R$ , it covers most single-element questions.

Networks reduce by two rules: series resistances add ( $R_{\text{eq}} = R_1 + R_2 + \cdots$ ) because the same current passes through each, and parallel resistances add reciprocally because each branch sees the same voltage. When a network will not reduce — for example, with multiple batteries — Kirchhoff's junction and loop rules give a complete system of equations.

Adding a capacitor makes the circuit time-dependent. A charging capacitor approaches its final charge as $1 - e^{-t/\tau}$ and a discharging capacitor decays as $e^{-t/\tau}$ , with $\tau = RC$ in both cases. After one time constant, about 63% of the change is complete; the exam usually tests the limits — a capacitor acts like a wire the instant switching occurs (if uncharged) and like an open circuit after a long time.

After this page, you should be able to

Apply Ohm's law to relate voltage, current, and resistance in a circuit element.
Reduce series and parallel resistor networks to an equivalent resistance.
Set up junction and loop equations for circuits that do not reduce to series and parallel blocks.
Describe RC charging and discharging as exponentials governed by the time constant.

Formulas and assumptions

Ohm's law

V = IR

Variables

V: voltage across the element in volts
I: current through the element in amperes
R: resistance in ohms

Assumptions

The element is ohmic: R does not depend on V or I.
Temperature effects on resistance are neglected.

Series resistors

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

Variables

R_eq: equivalent resistance in ohms
R1, R2, ...: individual resistances in ohms

Assumptions

The same current flows through every resistor in the chain.

Parallel resistors

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

Variables

R_eq: equivalent resistance in ohms
R1, R2, ...: individual resistances in ohms

Assumptions

Every resistor is connected across the same pair of nodes, so all share one voltage.

Electrical power

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

Variables

P: power dissipated in watts
I: current in amperes
V: voltage in volts
R: resistance in ohms

Assumptions

The three forms are equivalent only for a resistor obeying V = IR.

RC time constant

\tau = RC

Variables

tau: time constant in seconds
R: resistance in ohms
C: capacitance in farads

Assumptions

R is the resistance through which the capacitor charges or discharges.

Capacitor discharge

V(t) = V_0\, e^{-t/\tau}

Variables

V(t): capacitor voltage at time t in volts
V_0: initial voltage in volts
tau: time constant RC in seconds

Assumptions

The capacitor discharges through a resistor with no source in the loop.
Charging toward a final value follows 1 - e^(-t/tau) instead.

Worked example

Series-parallel network reduction

A $12\,\text{V}$ ideal battery drives a $4.0\,\Omega$ resistor in series with a parallel pair of $6.0\,\Omega$ and $3.0\,\Omega$ . Find the battery current and the current through each parallel resistor.

1Reduce the parallel pair: $R_{\text{par}} = \dfrac{(6.0)(3.0)}{6.0 + 3.0} = \dfrac{18}{9} = 2.0\,\Omega$ .
2Total resistance: $R_{\text{eq}} = 4.0 + 2.0 = 6.0\,\Omega$ , so the battery current is $I = 12/6.0 = 2.0\,\text{A}$ .
3Voltage across the parallel pair: $V_{\text{par}} = I R_{\text{par}} = (2.0)(2.0) = 4.0\,\text{V}$ (equivalently $12 - 2.0\times 4.0 = 4.0\,\text{V}$ ).
4Branch currents: $I_6 = 4.0/6.0 \approx 0.67\,\text{A}$ and $I_3 = 4.0/3.0 \approx 1.3\,\text{A}$ .
5Junction-rule check: $0.67 + 1.33 = 2.0\,\text{A}$ , matching the battery current.

The battery supplies $2.0\,\text{A}$ ; about $0.67\,\text{A}$ flows through the $6.0\,\Omega$ resistor and $1.3\,\text{A}$ through the $3.0\,\Omega$ resistor.

Common traps

Adding parallel resistances directly instead of through reciprocals, or forgetting to invert $1/R_{\text{eq}}$ at the end.
Assuming the full battery voltage appears across each element in a series chain rather than splitting in proportion to resistance.
Sign errors in loop equations from not fixing a consistent traversal direction before writing potential changes.
Treating a charging capacitor as if current flows 'through' it forever; after several time constants the branch current approaches zero.
Mixing up the charging form $1 - e^{-t/\tau}$ with the discharging form $e^{-t/\tau}$ .

GRE Physics overview Gauss's Law and Electric Fields Energy and Momentum Conservation Sign in for current practice

Question depth and domain coverage vary by exam. Practice answers are checked after submission.

Sources

GRE Subject Test Content and Structure — ETS. Accessed 2026-07-06. Use as a cited source for exam facts; do not imply affiliation or reproduce protected test material.
University Physics Volume 2, Section 9.4: Ohm's Law — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
University Physics Volume 2, Section 10.2: Resistors in Series and Parallel — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
University Physics Volume 2, Section 10.3: Kirchhoff's Rules — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
University Physics Volume 2, Section 10.5: RC Circuits — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.

Review and maintenance

Source-checked by publisher

Publisher record: Keiko Study editorial owner; Publisher placeholder; no individual credential claim is made yet.
Reviewer record: Technical reviewer pending; Placeholder only; this content is not labeled as reviewed by a named specialist.
Last source check: 2026-08-02
Next scheduled review: 2026-11-02

Recheck ETS content areas and the OpenStax current and circuits references before each major GRE Physics preparation cycle.

Report an issue with this page

DC Circuits

Concise answer

Definitions

Intuition

Concept walkthrough

After this page, you should be able to

Formulas and assumptions

Ohm's law

Series resistors

Parallel resistors

Electrical power

RC time constant

Capacitor discharge

Worked example

Series-parallel network reduction

Common traps

Related pages and practice

Sources

Review and maintenance