Physics 202 & 2B Lectures Notes

Resistance, Resistors, and Electric Circuits

Resistance

• Resistance is the measure of how much a material opposes the flow of electric current.
• Resistors are circuit elements designed to provide a specific resistance.
• Ohm’s Law: $V = IR$, where:
  • $V$ is the voltage across the resistor.
  • $I$ is the current flowing through the resistor.
  • $R$ is the resistance.
• Units of resistance: Ohms ($\Omega$).

Series Circuits

• Components are connected end-to-end, so the same current flows through each component.
• Equivalent resistance (Req) for resistors in series: $R<em>{eq} = R</em>1 + R_2 + …$
• The total voltage drop across series resistors is the sum of individual voltage drops.
• $I<em>1 = I</em>2 = I_{eq}$
• $V<em>1 + V</em>2 = V_{eq}$

Parallel Circuits

• Components are connected such that they provide multiple paths for current flow.
• The voltage across each component is the same.
• Equivalent resistance (Req) for resistors in parallel: $\frac{1}{R<em>{eq}} = \frac{1}{R</em>1} + \frac{1}{R_2} + …$
• The total current is the sum of the currents through each parallel branch.
• $V<em>1 = V</em>2 = V_{eq}$
• $I<em>1 + I</em>2 = I_{eq}$

Ohm's Law

• Defines the relationship between voltage (V), current (I), and resistance (R): $V = IR$
• If resistance is constant, voltage and current are directly proportional.
• Used to calculate voltage drop, current, or resistance in a circuit.

Electric Circuits

• A closed loop or path through which electric charge can flow.
• Requires a voltage source (e.g., battery) to drive the current.
• Components: resistors, capacitors, inductors, voltage sources, switches, etc.

Current

• The rate of flow of electric charge past a point or region.
• Formula: $I = \frac{\Delta Q}{\Delta t}$, where:
  • $\Delta Q$ is the amount of charge flowing.
  • $\Delta t$ is the time interval.
• Units: Ampere (A), where 1 A = 1 Coulomb/second.
• Conventional current: direction of positive charge flow (opposite to electron flow).

Kirchhoff’s Laws

• Kirchhoff's Current Law (Junction Rule):
  • The total current entering a junction (node) must equal the total current leaving the junction.
  • $\Sigma I<em>{in} = \Sigma I</em>{out}$
• Kirchhoff's Voltage Law (Loop Rule):
  • The sum of the voltage drops around any closed loop in a circuit must equal zero.
  • $\Sigma V = 0$
• Used to analyze complex circuits with multiple loops and junctions.

Conductors in Electrostatic Equilibrium

• Excess charge resides on the surface.
• The electric field inside is zero.
• The surface is an equipotential.
• The exterior electric field is perpendicular to the surface.
• The surface charge density and electric field strength are largest at sharp corners.

Capacitors

How Capacitors Work

• Capacitors store electrical energy in an electric field.
• When a voltage is applied, the capacitor charges, accumulating charge on its plates.
• The potential difference across the capacitor equals the battery voltage when fully charged.
• Electrons effectively move from one plate to the other.
• Capacitors can deliver energy faster than batteries but cannot hold as much energy for the same size.

Supercapacitors

• Used in some electric vehicles, trains, and buses.
• Advantages:
  • Rapid charging and discharging.
  • Longer lifetime and safer than batteries.
• Disadvantages:
  • Lower energy density compared to batteries.

Definition of Capacitance

• Capacitance (C): The ratio of charge (Q) to voltage (V).
• Formula: $C = \frac{Q}{V}$
• Units: Farad (F).
• Larger capacitance means more charge can be stored at a given voltage.

Parallel Plate Capacitor

• Capacitance depends on area (A) and separation (d) of plates.
• Formula: $C = \frac{\varepsilon_0 A}{d}$, where:
  • $\varepsilon<em>0$ is the permittivity of free space ($\varepsilon</em>0 \approx 8.85 \times 10^{-12} \frac{C^2}{Nm^2}$).
  • A is the area of one of the plates.
  • d is the distance between the plates.
• Closer plates and larger area increase capacitance.

Dielectrics in Capacitors

• Insulating material (dielectric) placed between capacitor plates.
• Allows closer plate spacing and higher capacitance without electron flow between plates.
• New capacitance formula: $C = \frac{\kappa \varepsilon_0 A}{d}$, where $\kappa$ is the dielectric constant.
• Dielectric material polarizes, creating an opposing electric field, reducing the overall electric field and increasing capacitance.

Capacitors in Series and Parallel

• Series:
  • Equivalent capacitance: $\frac{1}{C<em>{eq}} = \frac{1}{C</em>1} + \frac{1}{C_2} + …$
  • Charge is the same on all capacitors: $Q<em>1 = Q</em>2$
  • Voltage adds: $V<em>{eq} = V</em>1 + V_2$
• Parallel:
  • Equivalent capacitance: $C<em>{eq} = C</em>1 + C_2 + …$
  • Voltage is the same across all capacitors: $V<em>1 = V</em>2$
  • Charge adds: $Q<em>{eq} = Q</em>1 + Q_2$

Energy Stored in a Capacitor

• Energy (U) stored: $U = \frac{1}{2} QV = \frac{1}{2} CV^2 = \frac{1}{2} \frac{Q^2}{C}$

Spherical Shell Example with Gauss's Law

Problem Setup

• Nonconducting spherical shell with inner radius $R<em>1$ and outer radius $R</em>2$.
• Uniform volume charge density $\rho$ throughout the shell.

Gauss's Law Application $\Phi = \oint \vec{E} \cdot d\vec{A} = \frac{q<em>{enc}}{\varepsilon</em>0}$

Cases

• (a) r < R_1: Electric field is zero.
• (b) $R<em>1 < r < R</em>2$: Electric field is non-zero.
• (c) r > R_2: Electric field is non-zero.

Electric Field for r < R1

• Electric field: $E = 0$.
• Flux: $\Phi = 0$

Electric Field for R1 < r < R2

• Enclosed charge: $q<em>{enc} = \rho \cdot \frac{4}{3} \pi (r^3 - R</em>1^3)$

Electric Field for r > R2

• Enclosed charge: $q<em>{enc} = \rho \cdot \frac{4}{3} \pi (R</em>2^3 - R_1^3)$

Infinitely Long Charged Rod and Cylindrical Shell

Problem Setup

• Infinitely long conducting cylindrical rod with charge $\lambda$ per unit length.
• Surrounded by a conducting cylindrical shell with charge $-2\lambda$ per unit length and radius $r_1$.

Electric Field Between Rod and Shell

Surface Charge Densities

• Inner surface of the shell: $\sigma_{inner}$
• Outer surface of the shell: $\sigma_{outer}$

Electric Field Outside the Shell

Flux Calculation

• Cylindrical Gaussian surface between rod and shell.
• Flux through flat ends: 0.
• Flux through the wall of the cylinder: $\Phi_e = 2\pi rLE$

Electromotive Force (EMF)

• EMF is the maximum potential difference a battery or power source can provide.
• Denoted by $\varepsilon$.

Electrical Current

• The continuous flow of charge through a circuit.
• Current flows from the positive to the negative terminal (conventional current).
• Defined as the net amount of charge through a point per unit time: $I = \frac{\Delta Q}{\Delta t}$.
• Units: Ampere (A) = Coulomb/second.

Plinko Analogy

• Electrons flow through a conductor similar to disks bouncing through atoms in Plinko.
• Angle of incline represents EMF (potential difference).

Resistance

• Electrons bump into things while traveling through a conductor, slowing them down.
• Resistance (R) is determined by resistivity ($\rho$), length (L), and cross-sectional area (A): $R = \rho \frac{L}{A}$.
• Resistivity ($\rho$): Density of bumps in the conductor
• Length (L): Length of conductor
• A: Cross sectional area of conductor

Ohm’s Law

• Relates voltage, current, and resistance: $V = IR$
• Units of resistance: Ohms ($\Omega$).

Energy and Power

• The power dissipated by a resistor: $P = I^2R = \frac{V^2}{R} = IV$
• In a circuit, the resistance of connecting wires is generally small and negligible.

Kirchhoff’s Laws

• Kirchhoff’s Loop Law:
  • The sum of voltage drops around a closed loop is zero.
• Kirchhoff’s Junction Law:
  • The total current entering a junction equals the total current leaving it.

RC Circuits

Charging

• Capacitor starts as a wire (uncharged, Q = 0).
• Over time, the circuit blocks current (fully charged, $Q = Q_{max}$).
• Charge increases with time according to: $Q(t) = Q_{max} (1 - e^{-t/RC})$

Discharging

• Capacitor starts fully charged ($Q = Q_{max}$).
• Charge decreases to zero over time (Q = 0).
• Charge decreases with time according to: $Q = Q_{max} e^{-t/RC}$

Time Constant

• $\tau = RC$
• Characterizes the rate of charging or discharging.
• After one time constant, the charge decreases to approximately 36.8% of its initial value (discharging).
• In other words, in one time constant, the capacitor loses 63.2% of its initial charge
• Also represents the time required for the charge to increase from zero to 63.2% of its maximum

Clicker Question 1

• Smaller time constant $\tau = RC$