Electric current is the rate of charge flow:

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

Units: Amperes (A) = C/s

Conventional current:

• Defined as flow of positive charge
• Opposite direction to electron flow

Current density: $J = \frac{I}{A}$ (A/m2)

Drift velocity: $I = nqAv_d$

Where:

• n = charge carrier density
• q = charge per carrier
• A = cross-sectional area
• $v_d$ = drift velocity

Resistance: opposition to current flow

$R = \frac{V}{I}$

Units: Ohms (Ω) = V/A

Resistivity: material property

$R = \rho \frac{L}{A}$

Where:

• $\rho$ = resistivity (Ω - m)
• L = length
• A = cross-sectional area

Temperature dependence: $\rho = \rho_0[1 + \alpha(T - T_0)]$

Where $\alpha$ = temperature coefficient

Relationship between V, I, and R:

$V = IR$

Valid for ohmic materials:

• Linear I-V relationship
• R is constant

Non-ohmic devices:

• Diodes, transistors
• I-V not linear

Components connected end-to-end.

Properties:

• Same current through all components: $I_1 = I_2 = I$
• Voltages add: $V = V_1 + V_2 + ...$
• Equivalent resistance: $R_{eq} = R_1 + R_2 + ...$
• One component fails -> all fail

Voltage divider: $V_1 = \frac{R_1}{R_{eq}}V_{total}$

Current in series: $I = \frac{V_{total}}{R_{eq}}$

Components connected across same voltage.

Properties:

• Same voltage across all components: $V_1 = V_2 = V$
• Currents add: $I = I_1 + I_2 + ...$
• Equivalent resistance: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ...$
• One component fails -> others still work

Current divider: $I_1 = \frac{R_{eq}}{R_1}I_{total}$

Special case (two parallel resistors): $R_{eq} = \frac{R_1 R_2}{R_1 + R_2}$

Power Dissipation in Resistors:

Power (rate of energy dissipation):

$P = IV$

Using Ohm's law ($V = IR$):

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

Units: Watts (W) = Joules/second

Physical meaning:

• Energy converted to heat in resistor
• P = rate of electrical energy -> thermal energy

Energy dissipated: $E = Pt = IVt = I^2Rt$

Sum of currents entering junction equals sum leaving:

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

Or: $\sum I = 0$

Conservation of charge: charge cannot accumulate at junction.

Sum of voltages around any closed loop equals zero:

$\sum V = 0$

Physical meaning: electric field is conservative (work around closed loop = 0)

Sign convention:

• Voltage rise: positive (going from - to + across battery)
• Voltage drop: negative (going across resistor in direction of current)

Real batteries have internal resistance r.

Terminal voltage: $V_{terminal} = \mathcal{E} - Ir$

Where:

• $\mathcal{E}$ = emf (ideal voltage)
• I = current
• r = internal resistance

Circuit with internal resistance: $I = \frac{\mathcal{E}}{R_{external} + r}$

Power lost in battery: $P_{internal} = I^2r$

Capacitance measures ability to store charge:

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

Units: Farads (F) = C/V

Physical meaning: how much charge stored per volt

For parallel plates:

$C = \varepsilon_0\frac{A}{d}$

Where:

• $\varepsilon_0 = 8.85 \times 10^{-12}$ F/m (permittivity of free space)
• A = plate area
• d = separation

Factors affecting capacitance:

• Larger area A -> larger C
• Smaller separation d -> larger C
• Dielectric material between plates -> larger C

With dielectric: $C = \varepsilon_r\varepsilon_0\frac{A}{d} = \kappa\frac{A}{d}$

Where $\varepsilon_r$ = relative permittivity (dielectric constant)

Energy stored in electric field:

$U = \frac{1}{2}CV^2 = \frac{1}{2}\frac{Q^2}{C} = \frac{1}{2}QV$

Energy density: $u = \frac{U}{Volume} = \frac{1}{2}\varepsilon_0 E^2$

Work to charge capacitor: $W = \int_0^Q V \, dq = \int_0^Q \frac{q}{C} \, dq = \frac{Q^2}{2C}$

Dielectric: insulating material that can be polarized.

Effects:

• Increases capacitance by factor $\varepsilon_r$ (dielectric constant)
• Reduces electric field between plates
• Can withstand higher voltages before breakdown

Capacitance with dielectric: $C = \varepsilon_r C_0$

Electric field with dielectric: $E = \frac{E_0}{\varepsilon_r}$

Applications:

• Capacitors (store more energy)
• Cables (prevent breakdown)
• Circuit boards (insulation)

Steady state after long time ($t \gg \tau$):

• Capacitor fully charged
• Current through capacitor: $I_C = 0$
• Capacitor acts as open circuit
• Voltage across capacitor: $V_C = V_{applied}$

DC circuit behavior:

• Initially: capacitor uncharged, acts like short circuit
• Final state: capacitor charged, acts like open circuit
• Time constant $\tau = RC$ determines transition time

Charging (RC circuit with battery):

Charge as function of time: $q(t) = Q_{max}(1 - e^{-t/\tau})$

Voltage across capacitor: $V_C(t) = V_0(1 - e^{-t/\tau})$

Current: $I(t) = \frac{V_0}{R}e^{-t/\tau}$

Discharging (RC circuit without battery):

Charge: $q(t) = Q_0 e^{-t/\tau}$

Voltage: $V_C(t) = V_0 e^{-t/\tau}$

Current: $I(t) = -\frac{V_0}{R}e^{-t/\tau}$

Time constant: $\tau = RC$

At $t = \tau$: reaches ~63% of final value At $t = 5\tau$: essentially at steady state (~99%)

Kirchhoff's Junction Rule (Current Law):

Sum of currents entering junction equals sum leaving:

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

Or: $\sum I = 0$

Conservation of charge: charge cannot accumulate at junction.

Kirchhoff's Loop Rule (Voltage Law):

Sum of voltages around any closed loop equals zero:

$\sum V = 0$

Physical meaning: electric field is conservative (work around closed loop = 0)

Sign convention:

• Voltage rise: positive (going from - to + across battery)
• Voltage drop: negative (going across resistor in direction of current)