# Lecture 6: Electric Current and Resistance ## Physics II — Electromagnetism, Optics & Modern Physics --- ## Learning Objectives By the end of this lecture, students will be able to: 1. Define electric current and distinguish between conventional current and electron flow 2. State Ohm's law and identify ohmic and non-ohmic materials 3. Explain the microscopic model of current flow (drift velocity) 4. Calculate resistance from resistivity and geometric factors 5. Analyze power dissipation in resistive circuits --- ## Lecture Content ### I. Electric Current **Electric current** is the rate of flow of electric charge through a cross-section of a conductor: I = dQ/dt. For a steady current, this simplifies to I = Q/t, the total charge divided by the elapsed time. The SI unit of current is the **ampere** (A), equal to 1 C/s. **Conventional current** is defined as flowing in the direction that positive charges would move. In metals, the actual charge carriers are electrons, which flow opposite to the conventional current direction. By long-standing convention, circuit analysis uses the direction of positive charge flow. Current is technically a scalar quantity, though we assign it a direction of flow for circuit analysis. **Direct current (DC)** flows in one direction, while **alternating current (AC)** reverses direction periodically. ### II. Microscopic Model: Drift Velocity In a conductor without an applied electric field, free electrons move randomly at high thermal speeds of roughly 10^6 m/s, but their net displacement is zero because the motion is in all directions equally. When an electric field is applied, the electrons acquire a small net drift velocity v_d superimposed on their random thermal motion. This drift velocity is typically very small, on the order of 10^-4 m/s for typical currents in household wiring. The relationship between current and drift velocity is I = nAv_d, where n is the number density of charge carriers (electrons per unit volume), A is the cross-sectional area of the conductor, and v_d is the drift velocity. The **current density** J = I/A = nv_d has units of A/m^2. In vector form, J = n q v_d, pointing in the direction of conventional current. A natural question arises: if the drift velocity is so slow, why do lights turn on instantly when a switch is flipped? The answer is that the electric field propagates through the wire at nearly the speed of light. All electrons in the wire begin drifting almost simultaneously, even though each individual electron moves quite slowly. A cylindrical wire cross-section showing the microscopic picture of current flow. Random zigzag paths of individual electrons are shown (representing thermal motion), with a slight net drift to the left (electron flow) indicated by a dashed arrow labeled v_d. The conventional current direction I is shown with a bold arrow pointing to the right (opposite to electron drift). The cross-sectional area A is indicated, and the wire contains dots representing the free electrons with density n. Below, the equation I = nAv_d is displayed. ### III. Resistance and Ohm's Law **Ohm's law** states that the voltage across a conductor is proportional to the current through it: V = IR, where R is the **resistance** of the conductor. The SI unit of resistance is the **ohm** (Omega) = 1 V/A. **Resistivity** (rho) is an intrinsic property of the material itself, independent of the conductor's shape. The resistance of a conductor is related to its resistivity by R = rho L / A, where L is the length and A is the cross-sectional area. Resistivity is measured in ohm-meters (Omega m). Metals have very low resistivities of around 10^-8 Omega m, while insulators have extremely high resistivities ranging from 10^10 to 10^16 Omega m. The reciprocal of resistivity is **conductivity**: sigma = 1/rho, measured in 1/(Omega m) or S/m. The microscopic form of Ohm's law is J = sigma E = E/rho. Materials that obey Ohm's law, exhibiting a linear relationship between V and I (constant resistance), are called **ohmic materials**. Metals at constant temperature are good examples. **Non-ohmic materials** have resistance that varies with voltage or current. Diodes, light bulbs (whose filament temperature changes with current), and semiconductors are common examples. ### IV. Temperature Dependence of Resistance For most metals, resistivity increases approximately linearly with temperature: rho(T) = rho_0 [1 + alpha (T - T_0)], where alpha is the temperature coefficient of resistivity, measured in 1/K or 1/degrees C. Metals have a positive alpha, meaning their resistance increases with temperature. Semiconductors behave oppositely: their resistivity **decreases** with increasing temperature because more charge carriers are thermally excited into the conduction band, giving them an effectively negative temperature coefficient. **Superconductors** represent the extreme case, with resistivity dropping to exactly zero below a critical temperature T_c. Superconductors are used in MRI magnets, particle accelerators, and quantum computers. ### V. Electric Power The power delivered to any circuit element is P = IV. For a resistor, Ohm's law allows this to be rewritten as P = I^2 R = V^2 / R. The energy is converted to heat through a process called Joule heating. The SI unit of power is the watt (W). The kilowatt-hour (kWh) is a unit of energy commonly used in electrical billing: 1 kWh = 3.6 x 10^6 J. As a practical example, a 100 W light bulb operating at 120 V draws a current of I = P/V = 0.83 A and has a resistance of R = V^2/P = 144 Omega. Power transmission lines operate at high voltage specifically to reduce current and thereby minimize I^2R losses in the wires. A simple circuit with a battery (EMF = V) connected to a resistor R. Current I flows clockwise. Energy flow is illustrated: the battery converts chemical energy to electrical energy (P_battery = IV), and the resistor converts electrical energy to thermal energy (P_resistor = I^2R). Arrows show energy transformation at each component. Below, a comparison table shows the three equivalent power formulas: P = IV, P = I^2R, P = V^2/R, with notes on when each form is most useful. ### VI. EMF and Internal Resistance An **electromotive force (EMF)** source, such as a battery or generator, maintains a potential difference in a circuit. The EMF (epsilon) is defined as the work done per unit charge by the source, measured in volts. Despite its name, EMF is not actually a force. Real batteries have **internal resistance** (r), which causes the terminal voltage to be less than the EMF when current flows: V_terminal = epsilon - Ir. As the current increases, the terminal voltage decreases. The total power from the battery is P_total = epsilon I, which is split between the power delivered to the external circuit (P_external = I^2 R) and the power dissipated internally (P_internal = I^2 r). In a short circuit, where R = 0, the current reaches its maximum value of I = epsilon/r, and all power is dissipated internally. This is a dangerous condition that can cause overheating and damage. A circuit diagram showing a battery with EMF epsilon and internal resistance r (drawn as a small resistor inside the battery symbol) connected to an external resistance R. Current I = epsilon/(R + r) flows in the circuit. Voltage labels show: epsilon across the ideal EMF source, V_r = Ir across the internal resistance, and V_terminal = epsilon - Ir = IR across the external resistor. A voltmeter is shown connected across the battery terminals reading V_terminal.