Formula Sheet: Current and Resistance

Formula Sheet: Current and Resistance

Electric Current

Definition and Fundamental Formulas

• Electric Current: Rate of charge flow through a cross-sectional area \[I = \frac{Q}{t}\] where:
  • \(I\) = current (amperes, A)
  • \(Q\) = charge (coulombs, C)
  • \(t\) = time (seconds, s)
• Current in terms of charge carriers: \[I = nqAv_d\] where:
  • \(n\) = number density of charge carriers (carriers/m³)
  • \(q\) = charge per carrier (C)
  • \(A\) = cross-sectional area (m²)
  • \(v_d\) = drift velocity (m/s)
• Direction of current: By convention, current flows in the direction of positive charge movement (opposite to electron flow in conductors)

Resistance and Resistivity

Ohm's Law

• Ohm's Law: Relationship between voltage, current, and resistance \[V = IR\] where:
  • \(V\) = voltage (volts, V)
  • \(I\) = current (amperes, A)
  • \(R\) = resistance (ohms, Ω)
  Note: Applies to ohmic materials (materials with constant resistance)

Resistance of a Conductor

• Resistance formula: \[R = \frac{\rho L}{A}\] where:
  • \(R\) = resistance (Ω)
  • \(\rho\) = resistivity of material (Ω·m)
  • \(L\) = length of conductor (m)
  • \(A\) = cross-sectional area (m²)
  Key relationships:
  • Resistance increases with length
  • Resistance decreases with cross-sectional area
• Resistivity: Material property that quantifies how strongly a material opposes current flow
  • Units: Ω·m (ohm-meters)
  • Lower resistivity = better conductor
  • Higher resistivity = better insulator
• Conductivity: Reciprocal of resistivity \[\sigma = \frac{1}{\rho}\] where \(\sigma\) = conductivity (S/m or Ω^-1·m^-1)

Temperature Dependence of Resistance

• Temperature coefficient of resistance: \[R = R_0[1 + \alpha(T - T_0)]\] where:
  • \(R\) = resistance at temperature \(T\)
  • \(R_0\) = resistance at reference temperature \(T_0\)
  • \(\alpha\) = temperature coefficient of resistivity (K^-1 or °C^-1)
  • \(T\) = final temperature
  • \(T_0\) = reference temperature
  Note: For most metals, \(\alpha > 0\) (resistance increases with temperature)

Electrical Power

• Power dissipation (general): \[P = IV\] where:
  • \(P\) = power (watts, W)
  • \(I\) = current (A)
  • \(V\) = voltage (V)
• Power in a resistor (using Ohm's Law): \[P = I^2R\] \[P = \frac{V^2}{R}\] Applications: These formulas describe Joule heating or power dissipated as heat in a resistor
• Energy dissipated: \[E = Pt = IVt = I^2Rt = \frac{V^2t}{R}\] where:
  • \(E\) = energy (joules, J)
  • \(t\) = time (s)

Series and Parallel Circuits

Resistors in Series

• Equivalent resistance (series): \[R_{eq} = R_1 + R_2 + R_3 + ...\] Characteristics:
  • Same current through all resistors: \(I_{total} = I_1 = I_2 = I_3\)
  • Voltage divides across resistors: \(V_{total} = V_1 + V_2 + V_3 + ...\)
  • Equivalent resistance is greater than any individual resistance
• Voltage division in series: \[V_n = V_{total} \times \frac{R_n}{R_{eq}}\]

Resistors in Parallel

• Equivalent resistance (parallel): \[\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ...\] Characteristics:
  • Same voltage across all resistors: \(V_{total} = V_1 = V_2 = V_3\)
  • Current divides among branches: \(I_{total} = I_1 + I_2 + I_3 + ...\)
  • Equivalent resistance is less than the smallest individual resistance
• Special case (two resistors in parallel): \[R_{eq} = \frac{R_1 R_2}{R_1 + R_2}\]
• Current division in parallel: \[I_n = I_{total} \times \frac{R_{eq}}{R_n}\] Note: Higher current flows through lower resistance paths

Kirchhoff's Laws

Kirchhoff's Current Law (KCL)

• Junction Rule: The sum of currents entering a junction equals the sum of currents leaving \[\sum I_{in} = \sum I_{out}\] Alternative form: \[\sum I = 0\] where currents entering are positive and currents leaving are negative
  Basis: Conservation of charge

Kirchhoff's Voltage Law (KVL)

• Loop Rule: The sum of voltage changes around any closed loop is zero \[\sum V = 0\] Sign conventions:
  • Moving through a battery from - to +: positive voltage change
  • Moving through a battery from + to -: negative voltage change
  • Moving through a resistor in the direction of current: negative voltage change (-IR)
  • Moving through a resistor against the direction of current: positive voltage change (+IR)
  Basis: Conservation of energy

DC Circuits with Capacitors

Capacitors in Series

• Equivalent capacitance (series): \[\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + ...\] Note: Equivalent capacitance is less than the smallest individual capacitance
• Special case (two capacitors in series): \[C_{eq} = \frac{C_1 C_2}{C_1 + C_2}\]

Capacitors in Parallel

• Equivalent capacitance (parallel): \[C_{eq} = C_1 + C_2 + C_3 + ...\] Note: Equivalent capacitance is greater than any individual capacitance

RC Circuits

Charging a Capacitor

• Charge as a function of time: \[Q(t) = Q_{max}(1 - e^{-t/RC})\] where:
  • \(Q(t)\) = charge at time \(t\)
  • \(Q_{max}\) = maximum charge = \(C\varepsilon\)
  • \(\varepsilon\) = emf of battery
  • \(RC\) = time constant \(\tau\)
• Voltage across capacitor during charging: \[V_C(t) = \varepsilon(1 - e^{-t/RC})\]
• Current during charging: \[I(t) = \frac{\varepsilon}{R}e^{-t/RC}\] Note: Current is maximum at \(t = 0\) and decreases exponentially

Discharging a Capacitor

• Charge as a function of time: \[Q(t) = Q_0 e^{-t/RC}\] where \(Q_0\) = initial charge on capacitor
• Voltage across capacitor during discharging: \[V_C(t) = V_0 e^{-t/RC}\] where \(V_0\) = initial voltage
• Current during discharging: \[I(t) = I_0 e^{-t/RC}\] where \(I_0 = \frac{V_0}{R}\) Note: Current flows in opposite direction compared to charging

Time Constant

• RC Time Constant: \[\tau = RC\] where:
  • \(\tau\) = time constant (seconds)
  • \(R\) = resistance (Ω)
  • \(C\) = capacitance (F)
  Physical meaning:
  • Time for capacitor to charge to 63% of maximum (or discharge to 37% of initial)
  • After 5\(\tau\), capacitor is considered fully charged/discharged (≈99%)

Electromotive Force (EMF) and Internal Resistance

• Terminal voltage of a battery: \[V_{terminal} = \varepsilon - Ir\] where:
  • \(V_{terminal}\) = voltage across battery terminals (V)
  • \(\varepsilon\) = emf of battery (V)
  • \(I\) = current through battery (A)
  • \(r\) = internal resistance of battery (Ω)
  Note: Terminal voltage decreases as current increases
• Power delivered by battery: \[P_{delivered} = IV_{terminal} = I(\varepsilon - Ir)\]
• Power dissipated internally: \[P_{internal} = I^2r\]
• Total power supplied by emf: \[P_{total} = I\varepsilon\]

Electrical Measurements

Ammeters

• Purpose: Measure current
  • Connected in series with the circuit element
  • Ideal ammeter has zero resistance (no voltage drop)
  • Real ammeters have very low resistance

Voltmeters

• Purpose: Measure voltage
  • Connected in parallel with the circuit element
  • Ideal voltmeter has infinite resistance (draws no current)
  • Real voltmeters have very high resistance

Ohmmeters

• Purpose: Measure resistance
  • Circuit must be de-energized (no power)
  • Device provides its own current source

Additional Relationships and Concepts

Current Density

• Current density: \[J = \frac{I}{A}\] where:
  • \(J\) = current density (A/m²)
  • \(I\) = current (A)
  • \(A\) = cross-sectional area (m²)
• Microscopic Ohm's Law: \[J = \sigma E\] where:
  • \(\sigma\) = conductivity (Ω^-1·m^-1)
  • \(E\) = electric field (V/m)

Energy and Power in Batteries

• Energy stored in battery: \[E = Q\varepsilon\] where \(Q\) = total charge that can be delivered
• Efficiency of power transfer: \[\eta = \frac{P_{delivered}}{P_{total}} = \frac{V_{terminal}}{\varepsilon} = \frac{\varepsilon - Ir}{\varepsilon}\]
• Maximum power transfer theorem: Maximum power is delivered to load when load resistance equals internal resistance (\(R_{load} = r\))

Resistor Networks

• Delta-Wye (Δ-Y) transformations: Used for complex resistor networks (rarely tested on MCAT but useful for problem-solving)
• Simplification strategy:
  1. Identify series and parallel combinations
  2. Reduce step-by-step to equivalent resistance
  3. Apply Ohm's Law and Kirchhoff's Laws
  4. Work backwards to find individual currents and voltages

Important Constants and Values

• Elementary charge: \(e = 1.6 \times 10^{-19}\) C
• Typical resistivities (order of magnitude):
  • Conductors (copper): \(\rho \approx 10^{-8}\) Ω·m
  • Semiconductors (silicon): \(\rho \approx 10^{3}\) Ω·m
  • Insulators (glass): \(\rho \approx 10^{10}\) Ω·m or higher

Key Problem-Solving Tips

• For series circuits: Start with current (same everywhere), then find individual voltages
• For parallel circuits: Start with voltage (same everywhere), then find individual currents
• For complex circuits: Use Kirchhoff's Laws to set up simultaneous equations
• Sign conventions: Be consistent with direction of current and voltage drops
• Units: Always check units for consistency (Ω = V/A, W = V·A, etc.)
• RC circuits: At \(t = 0\), capacitor acts as wire (short circuit); at \(t = \infty\), capacitor acts as open circuit