Unit 2: Electric Circuits

Electric current is the rate of charge flow through a cross section:

By convention, current direction is the direction positive charge would flow — opposite to electron motion. Current density: \(\vec{J} = \sigma\vec{E}\) where \(\sigma\) is conductivity.

Resistance quantifies how much a material opposes current flow:

where \(\rho\) is resistivity (Ω·m), \(L\) is length, \(A\) is cross-sectional area. Resistivity depends on temperature: \(\rho = \rho_0[1 + \alpha(T - T_0)]\). For metals, \(\alpha > 0\) (resistance increases with heat); for semiconductors, \(\alpha < 0\).

Ohm's Law: \(V = IR\) applies to ohmic materials where R is independent of V. Non-ohmic devices (diodes, filaments) have a nonlinear I-V curve.

Electric Power:

\(I^2R\) is the Joule heating rate — energy lost as heat in a resistor.

Internal resistance (r) of real batteries: A real battery has internal resistance r in series with its ideal EMF ε. Terminal voltage: \(V_{\text{terminal}} = \varepsilon - Ir\). When current flows, terminal voltage is LESS than the EMF. Open circuit: \(I=0 \Rightarrow V_{\text{terminal}} = \varepsilon\). Short circuit: \(I_{\text{max}} = \varepsilon / r\).

In a series circuit, components share one path — current is the same through every element.

Equivalent resistance:

Voltage divider: The voltage across each resistor is proportional to its resistance:

Key facts:

• \(I = I_1 = I_2 = \cdots\) (same current everywhere)
• \(V_{\text{total}} = V_1 + V_2 + \cdots\) (voltages add)
• Largest resistor gets largest voltage drop (and dissipates most power)

Kirchhoff's Laws

Kirchhoff's Junction Rule (KCL): At any junction, \(\sum I_{\text{in}} = \sum I_{\text{out}}\) — charge is conserved.

Kirchhoff's Loop Rule (KVL): Around any closed loop, \(\sum \Delta V_i = 0\) — energy is conserved. The sum of all voltage rises (EMFs) equals the sum of all voltage drops (IR).

Sign conventions for loop rule: Going across a resistor in the direction of current: −IR. Going across a battery from − to +: +ε. From + to −: −ε.

Application strategy: (1) Label all currents. (2) Write junction equations. (3) Write loop equations for independent loops. (4) Solve the system.

In a parallel circuit, components share two nodes — voltage is the same across every branch.

Equivalent resistance:

For two resistors: \(R_{\text{eq}} = \frac{R_1 R_2}{R_1 + R_2}\)

Current divider: In a two-resistor parallel branch:

Key facts:

• \(V = V_1 = V_2 = \cdots\) (same voltage across each branch)
• \(I_{\text{total}} = I_1 + I_2 + \cdots\) (currents add)
• \(R_{\text{eq}} < \min(R_1, R_2, \dots)\) — adding a parallel path always decreases R_eq
• Smallest resistor carries largest current (and dissipates most power)

After a long time (all transients have decayed), a capacitor reaches steady state:

• Capacitor acts as an open circuit — no current flows through its branch
• Voltage across capacitor = voltage of its branch: \(V_C = V_{\text{branch}}\)
• Charge on capacitor: \(Q = C V_C\)
• Energy stored: \(U = \frac{1}{2}C V_C^2\)

Analysis strategy:

1. Replace all capacitors with open circuits
2. Solve the resulting resistor-network for voltages at capacitor positions
3. Then \(Q = CV\) and \(U = \frac{1}{2}CV^2\) for each capacitor

During charging/discharging, the capacitor voltage and current are exponential functions of time.

Time constant:

After \(1\tau\): ~63% of change. After \(5\tau\): effectively at steady state (>99%).

Charging (RC series, battery ε):

Discharging (no battery):

Standard derivative forms (memorize):

• Charging: \(\frac{dq}{dt} = \frac{\varepsilon}{R}e^{-t/RC}\)
• At \(t=0\): uncharged capacitor acts like a short (zero voltage)
• At \(t \to \infty\): capacitor acts like an open (no current)

Ammeters measure current. They must be connected in series and have very low internal resistance (ideally zero) so they don't affect the circuit.

Voltmeters measure potential difference. They must be connected in parallel across the component and have very high internal resistance (ideally infinite) so they draw negligible current.