E&M — Complete Constants, Equations & Unit Intuitions

Current & Drift Velocity

Definition Electric Current — all forms

$$I = \frac{dq}{dt} \qquad \Longleftrightarrow \qquad q = \int I\,dt$$ $$I = nqv_d A$$

Symbol	Unit	Dimensions	Meaning
$I$	A (ampere)	C/s	Charge flowing past a cross-section per second
$n$	m⁻³	1/m³	Number density of charge carriers (electrons per cubic meter)
$q$	C	A·s	Charge per carrier (for electrons: $-e$, but we use magnitude)
$v_d$	m/s	m/s	Drift velocity — the slow net motion of electrons. In copper at 1A through 1mm² wire, $v_d \approx 0.1$ mm/s! Electrons move slowly; the field signal travels at ~$c$.
$A$	m²	m²	Cross-sectional area of the conductor

What is an Ampere?

1 A = 1 C/s = 6.24 × 10¹⁸ electrons/s

One ampere is one coulomb per second. The household circuit breaker trips at 15-20A. An LED draws ~20 mA. Lightning is ~20,000 A for a few microseconds. A phone charger is 1–3 A. The key insight: a 1A current in a wire doesn't mean electrons are sprinting. They drift at mm/s. But there are so many electrons ($n \sim 10^{28}/\text{m}^3$ in copper) that even a tiny drift velocity carries 1 coulomb per second past any cross-section.

Density Current Density $\vec{J}$

$$\vec{J} = \frac{I}{A}\hat{\ell} = nq\vec{v}_d \qquad \text{and} \qquad \vec{J} = \sigma\vec{E}$$

Symbol	Unit	Meaning
$J$	A/m²	Current per unit area — tells you how "dense" the current flow is, not just total amps
$\sigma$	S/m = 1/(Ω·m)	Electrical conductivity — inverse of resistivity. High $\sigma$ = easy current flow = metal. Low $\sigma$ = insulator.

Resistance & Ohm's Law

Ohm's Law Ohm's Law — all forms & variations

$$V = IR \qquad \Longleftrightarrow \qquad I = \frac{V}{R} \qquad \Longleftrightarrow \qquad R = \frac{V}{I}$$

solve for $V$

$$V = IR$$

solve for $I$

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

solve for $R$

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

Symbol	Unit	Dimensions	Meaning
$V$	V (volt)	J/C	Potential difference across the resistor — the "pressure" driving current
$I$	A (ampere)	C/s	Current through the resistor — the "flow rate"
$R$	Ω (ohm)	V/A = kg·m²/(A²·s³)	Resistance — the "friction" or "narrowness" opposing current flow

What is an Ohm?

1 Ω = 1 V/A = 1 J·s/C² = kg·m²/(A²·s³)

V/A reading: "It takes 1 volt of push to drive 1 ampere through 1 ohm." A higher ohm means more voltage needed per amp of current.

Typical values: Household wire: ~0.01 Ω/m. Light bulb filament: 240 Ω (hot). Human body: 1,000–100,000 Ω (varies with moisture). A dry resistor in your lab kit: 100–10,000 Ω. A short circuit: ideally 0 Ω. An open circuit: ideally ∞ Ω.

Resistivity Resistance from Geometry & Material

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

solve for $\rho$

$$\rho = \frac{RA}{L}$$

solve for $L$

$$L = \frac{RA}{\rho}$$

temperature dependence

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

Symbol	Unit	Meaning
$\rho$	Ω·m	Resistivity — intrinsic material property. Copper: $1.7\times10^{-8}\ \Omega\cdot\text{m}$. Silicon: $\sim10^{3}$. Glass: $\sim10^{12}$. A 15-order-of-magnitude range!
$L$	m	Length of conductor — longer wire = more collisions = higher resistance
$A$	m²	Cross-sectional area — fatter wire = more parallel paths = lower resistance
$\alpha$	1/°C or 1/K	Temperature coefficient of resistivity — metals get more resistive when hot (more atomic vibrations to scatter electrons)

Combinations Resistors in Series and Parallel

Series

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

Same $I$, voltages add.
$R_\text{eq} > $ any individual $R$

Parallel

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

Same $V$, currents add.
$R_\text{eq} < $ any individual $R$

Two parallel resistors

$$R_\text{eq} = \frac{R_1 R_2}{R_1 + R_2}$$

Product-over-sum formula — shortcut for exactly two

Power & Energy in Circuits

Power Electric Power — all three forms

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

use when $I$ and $V$ known

$$P = IV$$

use when $I$ and $R$ known

$$P = I^2R$$

use when $V$ and $R$ known

$$P = V^2/R$$

Symbol	Unit	Dimensions	Meaning
$P$	W (watt)	J/s = kg·m²/s³	Rate of energy delivery or dissipation — joules per second
$IV$	A·V = W	(C/s)·(J/C) = J/s	Current times voltage — amps of flow times joules per coulomb = joules per second
$I^2R$	A²·Ω = W	(C/s)²·(V/A) = J/s	Joule heating — power dissipated as heat. Bigger current or bigger resistance = more heat. This is why high-voltage transmission lines use high $V$ and low $I$ — to minimize $I^2R$ losses.

What is a Watt?

1 W = 1 J/s = 1 V·A = kg·m²/s³

A watt is a joule per second — the rate of energy use, not the total energy. Your phone charger is 5–20 W. A microwave is 1000 W. A human in moderate exercise: ~100 W. A car engine: ~100,000 W (100 kW). The sun's output: $3.8\times10^{26}$ W.

Your electricity bill charges you in kilowatt-hours (kWh): $1\ \text{kWh} = 1000\ \text{W} \times 3600\ \text{s} = 3.6 \times 10^6\ \text{J} = 3.6\ \text{MJ}$. Running a 1000 W microwave for 1 hour uses 1 kWh.

EMF EMF and Terminal Voltage of a Battery

$$V_\text{terminal} = \mathcal{E} - Ir \qquad \Longleftrightarrow \qquad \mathcal{E} = V_\text{terminal} + Ir$$ $$P_\text{delivered} = \mathcal{E}I - I^2r = IV_\text{terminal}$$

Symbol	Unit	Meaning
$\mathcal{E}$	V	EMF (electromotive force) — the open-circuit voltage the battery can supply; energy per unit charge the battery's chemistry provides
$r$	Ω	Internal resistance of the battery — causes terminal voltage to sag under load
$Ir$	V	Voltage drop inside the battery itself — wasted energy heating the battery
$V_\text{terminal}$	V	What you actually measure at the battery terminals while current flows

Kirchhoff's Rules

KCL Kirchhoff's Current Law — Junction Rule

$$\sum_\text{junction} I = 0 \qquad \Longleftrightarrow \qquad \sum I_\text{in} = \sum I_\text{out}$$

Deep meaning: Conservation of charge. In steady state (DC), charge cannot accumulate at a node. Every coulomb that arrives must leave. Units: every term is in amperes [A = C/s]. The equation says charge flow rates balance.

How to apply KCL

1. Label every branch current (direction can be assumed — a negative result just means your assumed direction was backward).
2. At each junction: sum currents entering = sum currents leaving.
3. $N$ junctions give $N-1$ independent KCL equations (the last is redundant).

KVL Kirchhoff's Voltage Law — Loop Rule

$$\sum_\text{closed loop} \Delta V = 0$$

Deep meaning: Conservation of energy. The electric potential is a well-defined single-valued function. If you walk around any closed path and return to start, the total voltage change must be zero — you can't gain energy for free. Units: every term is in volts [V = J/C]. The equation says energy per unit charge is conserved per loop.

Element	Traversed with current	Traversed against current
Resistor $R$	$-IR$ (voltage drop)	$+IR$ (voltage rise)
Battery $\mathcal{E}$	$+\mathcal{E}$ (from − to + terminal)	$-\mathcal{E}$ (from + to − terminal)
Internal resistance $r$	$-Ir$	$+Ir$
Capacitor $C$	$-Q/C$ (drop toward + plate)	$+Q/C$

How to apply KVL

1. Choose a loop direction (clockwise or counterclockwise — doesn't matter, just be consistent).
2. Walk around the loop, writing $\Delta V$ for each element using the sign table above.
3. Set the sum to zero. Each independent loop gives one KVL equation.
4. $B$ branches − $N$ nodes + 1 = number of independent KVL loops (mesh analysis).

Solving a circuit — the systematic approach

Circuit-Solving Recipe

For a circuit with $B$ branches, $N$ nodes, and $L$ loops:

Assign a current variable and direction to each branch.
Write $(N-1)$ independent KCL equations at junctions.
Write $L = B - N + 1$ independent KVL equations around meshes.
You now have $B$ equations in $B$ unknowns — solve the linear system.
Negative current means your assumed direction was reversed — just flip the arrow.

This is the same matrix equation $A\vec{x} = \vec{b}$ you solve in linear algebra. KCL and KVL together form a complete, solvable system — that's the power of Kirchhoff.

All Units — Intuition Guide

Every SI unit in E&M can be expressed in base SI units: kg, m, s, A. Here's the complete translation:

Unit	Symbol	Equals	Base SI	Physical intuition
Coulomb	C	A·s	A·s	Amount of charge. 1 C = 6.24×10¹⁸ elementary charges. Enormous — practical currents involve fractions of a coulomb per second.
Volt	V	J/C	kg·m²/(A·s³)	Energy per unit charge. "Height" in the electrical landscape. The pressure that drives current. 9V battery gives each coulomb 9 joules.
Ampere	A	C/s	A (base unit)	Charge flow rate. 1A = 6.24×10¹⁸ electrons/second past a cross-section. Your phone charger: ~2A.
Ohm	Ω	V/A	kg·m²/(A²·s³)	Opposition to current flow. Ratio of voltage applied to current produced. Higher Ω = more voltage needed per amp.
Watt	W	J/s = V·A	kg·m²/s³	Rate of energy transfer. Your body at rest: ~80W. A 100W bulb: 100 joules per second converted to heat and light.
Farad	F	C/V	A²·s⁴/(kg·m²)	Charge stored per volt. 1F is huge — stores 1 coulomb per volt. Most circuits use µF or pF. Supercapacitors: 1–3000 F.
N/C = V/m	E field	N/C = V/m	kg·m/(A·s³)	Force per coulomb (N/C) = voltage gradient (V/m). Same quantity, two ways to think about it. Near a Van de Graaff at 100kV in 10cm: E = 10⁶ V/m = 1 MV/m.
N·m²/C	Φ_E	Electric flux	kg·m³/(A·s³)	Field strength × area. Counts how many field lines pierce through a surface. Used in Gauss's law.
J/m³	u	Energy density	kg/(m·s²)	Energy stored per cubic meter of field. The electric field itself carries energy — this is the energy density formula $u = \frac{1}{2}\varepsilon_0 E^2$.
Ω·m	ρ	Resistivity	kg·m³/(A²·s³)	Intrinsic resistance of a material regardless of geometry. Copper: 1.7×10⁻⁸ Ω·m. Glass: ~10¹² Ω·m. 20 orders of magnitude between best conductor and insulator.
A/m²	J	Current density	A/m²	Current per unit cross-sectional area. Safe limit for copper wire: ~10⁷ A/m². Fuses blow when current density exceeds the wire's thermal limit.
eV	eV	1.602×10⁻¹⁹ J	kg·m²/s²	Energy gained by one elementary charge across 1V. Atomic bond energies: 1–10 eV. X-ray photons: 10³–10⁶ eV. This unit is natural at atomic scales where joules are absurdly large.

The Ultimate Dimensional Sanity Check

When you're unsure if an equation is right, check units. Every valid physics equation must have the same dimensions on both sides. Here's the most useful chain to memorize:

J= N·m= C·V= A·V·s= W·s= kg·m²/s²

V= J/C= W/A= Ω·A= N·m/C= kg·m²/(A·s³)

If both sides of your equation have the same unit expression in base SI, the equation is dimensionally consistent. It might still be wrong (missing a factor of 2, wrong sign, etc.), but it can't be fundamentally broken.