NCEA Level 3 Physics: Electrical Systems Exhaustive Study Guide

Direct Current and Voltage Fundamentals

• Definition of Current ($I$): Current is the flow rate of charge, measured in Amperes ($A$). It is mathematically defined as:     $I = \frac{Q}{t}$

  • $I$: Current ($A$)

  • $Q$: Charge ($C$)

  • $t$: Time ($s$)

• Direction of Flow:

  • Conventional Current: Defined as the direction of positive charge flow.

  • Electron Flow: Negative electrons flow in the opposite direction of conventional current.

• Propagation Speeds:

  • Changes in current (electric field propagation) travel at approximately $3 \times 10^8\,m/s$.

  • Individual electrons themselves move very slowly, traveling at drift speeds of only millimeters per minute.

  • Analogy (Hosepipe): Turning on a tap with a hosepipe already full of water results in flow from the end almost immediately, even though a specific water particle may take several seconds to travel through the entire length of the hose.

• Voltage ($V$): Voltage is the energy lost or gained per charge between two points in an electric field, measured in Volts ($V$).     $V = \frac{E}{Q}$

  • $E$: Energy ($J$)

  • $Q$: Charge ($C$)

  • Practical Example: A lamp with a voltage of $12\,V$ across it means every Coulomb ($C$) of charge passing through it delivers $12\,J$ of energy.

Ohm’s Law and Circuit Resistance

• Resistance ($R$): As charges move through a conductor, they encounter resistance depending on the shape and structure of the material. It is measured in Ohms ($\Omega$).

• Ohm's Law: The relationship between voltage, current, and resistance is expressed as:     $V = IR$

  • Example: If $V = 12\,V$ and $I = 2\,A$, the resistance is $6\,\Omega$ because $12\,V = 2\,A \times 6\,\Omega$.

• Equivalent Resistance Configuration:

  • Series Resistance: Adding resistors increases total resistance.         $R_T = R_1 + R_2 + R_3$

  • Parallel Resistance: Adding resistors decreases total resistance.         $\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$

Source Resistance and Electromotive Force (EMF)

• Internal Resistance ($r$): Voltage sources contribute resistance to a circuit which can be represented as an electromotive force (emf, $\epsilon$ ) in series with an internal resistor ($r$).

• Terminal Voltage ($V$): The actual voltage available at the source terminals depends on the current flow:     $V = \epsilon - Ir$

  • $V$: Terminal voltage ($V$)

  • $\epsilon$: Electromotive force ($V$)

  • $I$: Current ($A$)

  • $r$: Internal resistance ($\Omega$)

• Worked Example: A battery with a $12\,V$ emf produces $0.5\,A$ when connected to a $20\,\Omega$ lamp.

  • $V = IR = 0.5\,A \times 20\,\Omega = 10\,V$

  • $V = \epsilon - Ir \rightarrow -Ir = V - \epsilon \rightarrow r = \frac{V - \epsilon}{-I}$

  • $r = \frac{10\,V - 12\,V}{-0.5\,A} = 4\,\Omega$

Kirchhoff’s Circuit Laws

• Kirchhoff's Current Law (KCL): The total current entering a node must equal the total current exiting that node.     $I_{in} = I_{out}$

  • Example: A node with inputs of $12\,A$ and $3\,A$ ($15\,A$ total) must have outputs totaling $15\,A$ (e.g., $7\,A$ and $8\,A$).

• Kirchhoff's Voltage Law (KVL): All voltages in a circuit loop sum to zero. The sum of potential gains (low to high energy) equals the sum of potential drops (high to low energy).

  • Following a Loop: Identify high and low energy sides of components. Gains are positive; drops are negative.

  • Example Loop 1: $12\,V$ source, $5\,V$ component, $7\,V$ component: $+12\,V - 5\,V - 7\,V = 0$.

  • Example Loop 2: $-4\,V - 5\,V + 9\,V = 0$.

Capacitors and Dielectrics

• Function: Capacitors store energy in electric fields (unlike batteries, which use chemical energy). Capacitor voltage is directly proportional to the stored charge ($Q \propto V$).

• Capacitance ($C$): Measured in Farads ($F$), it is the charge stored per volt applied.     $C = \frac{Q}{V}$

• Dielectrics: These are electric insulators allowing charge separation within themselves to store energy. They increase capacitance by physically separating plates. Effectiveness is measured by relative permittivity ($\epsilon_r$).

  • Vacuum/Air: $\epsilon_r = 1$

  • Paper: $\epsilon_r = 1.4$ to $3.50$

  • Glass: $\epsilon_r = 5$ to $7$

  • Water: $\epsilon_r = 80$

• Physical Factors affecting Capacitance:     $C = \frac{\epsilon_0 \epsilon_r A}{d}$

  • $\epsilon_0$: Vacuum permittivity ($8.85 \times 10^{-12}\,Fm^{-1}$)

  • $A$: Plate Area ($m^2$). Increasing area increases $C$.

  • $d$: Plate Separation ($m$). Decreasing separation increases $C$.

• Comparison: Changing Separation ($d$):

  • Connected to Voltage Source: $V$ remains constant. Increasing $d$ decreases $C$ ($C = \frac{\epsilon A}{d}$) and decreases $Q$ ($Q = CV$).

  • Disconnected from Source: $Q$ remains constant. Increasing $d$ decreases $C$, and therefore $V$ must increase ($V = \frac{Q}{C}$).

Capacitor Circuits and Energy

• Equivalent Capacitance:

  • Series: $\frac{1}{C_T} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3}$. Charge ($Q$) on each is equal.

  • Parallel: $C_T = C_1 + C_2 + C_3$. Voltage ($V$) across each is equal.

• Energy Stored in Capacitor ($E_p$):     $E = \frac{1}{2}QV = \frac{1}{2}CV^2$

  • Energy Losses: When charging, half the total energy supplied by the source ($E_{total} = QV$) is lost to resistance in the circuit. This is because current (and resistor voltage) decreases over time as the capacitor charges.

• RC Time Constant ($\tau$):     Charging and discharging follow an exponential curve. The time constant ($\tau$) is the time for a $63\%$ change.     $\tau = RC$

  • Charging Rule:

    • After $1\tau$: Value reaches $63\%$ of maximum.

    • After $5\tau$: Value reaches $99.33\%$ (treated as fully charged/$100\%$).

  • Discharging Rule: After $1\tau$, value decreases to $37\%$ of maximum ($100\% - 63\% = 37\%$).

Magnetic Flux and Faraday’s Law

• Magnetic Flux ($\phi$): The amount of magnetic field passing through a surface. Units: Webers ($Wb$).     $\phi = BA\cos(\theta)$

  • $B$: Magnetic field strength ($T$)

  • $A$: Area ($m^2$)

  • $\theta$: Angle to the perpendicular. Maximum flux occurs when the field is perpendicular to the area ($\cos(0) = 1$).

• Faraday's Law: The induced electromotive force (emf) is proportional to the rate of change of magnetic flux.     $\epsilon = -\frac{\Delta \phi}{\Delta t}$

  • The negative sign indicates the induced emf opposes the change that created it.

• Lenz's Law: When a solenoid experiences a changing magnetic field, the induced current creates a field opposing the change.

  • North towards: Induced North pole (Anticlockwise current).

  • North away: Induced South pole (Clockwise current).

  • South towards: Induced South pole (Clockwise current).

  • South away: Induced North pole (Anticlockwise current).

  • The effect is only present while the flux is actively changing (magnet in motion).

Inductors and Self-Inductance

• Self-Inductance ($L$): A measure of how effectively a conductor induces a "back emf" in itself due to a change in its own current. Measured in Henrys ($H$).     $\epsilon = -L\frac{\Delta I}{\Delta t}$

• Physical Inductor Factors: Inductance is increased by:

  • More coils: More interaction with flux.

  • More area: Requires more coil, leading to more flux interaction.

  • Tighter coils: Coils affect each other more closely.

  • Adding a core: (e.g., iron) allows flux to conduct with fewer losses.

• Inductor Energy ($E_p$): Energy is stored in the magnetic field while current flows.     $E_p = \frac{1}{2}LI^2$

• Inductor Time Constant ($\tau$):     $\tau = \frac{L}{R}$

  • Current and voltage follow an exponential curve similar to capacitors.

  • Charging: Current increases rapidly at first, causing a large opposing voltage.

• Application: Spark Plug:

  • Flux perspective: Opening a switch causes current to drop rapidly. This rapid change in flux generates a large emf, especially with many secondary turns, enough to jump the spark gap.

  • Time Constant perspective: The air in the spark gap is a huge resistance ($R$). Since $\tau = \frac{L}{R}$, a large $R$ makes $\tau$ very small. Current drops nearly instantly ($\Delta I/\Delta t$ is huge), producing a massive emf via $\epsilon = -L\frac{\Delta I}{\Delta t}$.

Transformers

• Mutual Inductor Mechanism: Two inductors arranged so flux change in one (primary) produces a change in the other (secondary). Uses laminated iron cores to reduce energy losses from eddy currents.

• Turn Ratio Formula:     $\frac{N_p}{N_s} = \frac{V_p}{V_s} = \frac{I_s}{I_p}$

  • $N$: Number of turns

  • $V$: Voltage ($V$)

  • $I$: Current ($A$)

• Efficiency:

  • Ideal: $P_{in} = P_{out}$

  • Real: $Efficiency = \frac{P_{out}}{P_{in}} \times 100\%$

Alternating Current (AC) Systems

• Sinusoidal Nature: Current and voltage change direction periodically.     $V = V_{max}\sin(\omega t)$     $I = I_{max}\sin(\omega t)$

  • $\omega$: Angular frequency ($rad\,s^{-1}$). $\omega = 2\pi f = \frac{2\pi}{T}$.

• Root Mean Square (RMS): Used to describe the effective average of the changing AC values.     $V_{max} = \sqrt{2}V_{RMS}$     $I_{max} = \sqrt{2}I_{RMS}$

• Multiplying $V_{RMS}$ and $I_{RMS}$ yields the same power value as an equivalent DC circuit.

Reactance and Impedance

• Reactance ($X$): Opposition to AC current where energy is stored rather than lost (measured in $\Omega$).

  • Capacitor Reactance ($X_C$): $X_C = \frac{1}{\omega C}$. (Decreases as frequency increases).

  • Inductor Reactance ($X_L$): $X_L = \omega L$. (Increases as frequency increases).

• Impedance ($Z$): Combines resistance ($R$) and reactance ($X$).     $Z = \sqrt{R^2 + X^2} = \sqrt{R^2 + (X_L - X_C)^2}$     $V = IZ$

• Phase Relationships:

  • $V_L$ leads $V_R$ by $90^\circ$.

  • $V_C$ lags $V_R$ by $90^\circ$.

  • $V_R$ is in phase with $I$.

LCR Resonance

• Resonant Frequency ($f_0$): The specific frequency where inductor reactance equals capacitor reactance ($X_L = X_C$).     $f_0 = \frac{1}{2\pi \sqrt{LC}}$

  • Since $X_L$ and $X_C$ are $180^\circ$ out of phase, they cancel entirely at resonance.

  • Consequences: Total reactance is zero. Impedance ($Z$) is at its minimum (equal to $R$). Current ($I$) is at its maximum.