🎢Principles of Physics II Unit 5 – DC Circuits

Key Concepts and Definitions

• Electric current $I$ represents the flow of electric charge through a conductor measured in amperes (A)
• Voltage $V$ signifies the potential difference between two points in a circuit measured in volts (V)
  • Voltage sources provide the energy needed to drive current through a circuit
• Resistance $R$ quantifies the opposition to current flow in a conductor measured in ohms ($\Omega$)
  • Conductors have low resistance allowing current to flow easily while insulators have high resistance impeding current flow
• Power $P$ denotes the rate at which electrical energy is converted into other forms (heat, light, etc.) measured in watts (W)
• Nodes are points in a circuit where two or more components connect
• Branches are paths between nodes through which current can flow
• Closed circuit enables current to flow from the voltage source through the components and back to the source
• Open circuit prevents current from flowing due to a break in the path

Circuit Components and Symbols

• Resistors limit current flow and have the symbol \boxed{\text{\sim}} with resistance value labeled
• Voltage sources provide energy to the circuit and have the symbol \boxed{\text{+}\atop\text{-}} for DC sources
  • Ideal voltage sources maintain a constant voltage regardless of the current drawn
• Current sources supply a constant current to the circuit and have the symbol \boxed{\text{\rightarrow}\atop\text{\bullet}}
• Wires are assumed to have negligible resistance and are represented by lines connecting components
• Switches control the flow of current and are shown as a line with a break \boxed{\text{\diagup}\atop\text{\diagdown}} (open) or a continuous line \boxed{\text{/}} (closed)
• Capacitors store electric charge and have the symbol \boxed{\text{||}} with capacitance value labeled in farads (F)
• Inductors store energy in a magnetic field and have the symbol \boxed{\text{\infty}} with inductance value labeled in henries (H)
• Ground is a reference point often used as the zero voltage level and has the symbol \boxed{\text{\perp}\atop\text{\diagdown}\atop\text{\diagup}}

Ohm's Law and Resistance

• Ohm's law states that voltage $V$ equals current $I$ times resistance $R$: $V = IR$
  • Doubling the voltage across a resistor doubles the current through it if resistance remains constant
  • Halving the resistance of a component doubles the current through it if voltage remains constant
• Resistance depends on the material's resistivity $\rho$, length $L$, and cross-sectional area $A$: $R = \frac{\rho L}{A}$
  • Longer wires have higher resistance while thicker wires have lower resistance for the same material
• Conductivity $\sigma$ is the reciprocal of resistivity: $\sigma = \frac{1}{\rho}$
• Temperature affects resistance with most materials having higher resistance at higher temperatures
  • Semiconductors are an exception and have lower resistance at higher temperatures
• Voltage dividers consist of resistors in series and output a voltage proportional to the input voltage based on the resistor values

Series and Parallel Circuits

• Series circuits have components connected end-to-end forming a single path for current
  • Current is the same through all components in series
  • Total voltage equals the sum of voltages across individual components: $V_{total} = V_1 + V_2 + \ldots$
  • Equivalent resistance equals the sum of individual resistances: $R_{eq} = R_1 + R_2 + \ldots$
• Parallel circuits have components connected across the same two nodes forming multiple paths for current
  • Voltage is the same across all components in parallel
  • Total current equals the sum of currents through individual components: $I_{total} = I_1 + I_2 + \ldots$
  • Reciprocal of equivalent resistance equals the sum of reciprocals of individual resistances: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots$
• Series-parallel circuits contain both series and parallel connections and can be analyzed by combining equivalent resistances

Kirchhoff's Laws

• Kirchhoff's current law (KCL) states that the sum of currents entering a node equals the sum of currents leaving the node: $\sum I_{in} = \sum I_{out}$
  • Conservation of charge implies that charge cannot accumulate at a node in a circuit at steady state
• Kirchhoff's voltage law (KVL) states that the sum of voltages around any closed loop in a circuit equals zero: $\sum V = 0$
  • Voltage rises are considered positive while voltage drops are considered negative
  • Conservation of energy implies that the net energy change around a closed loop must be zero
• Applying KCL and KVL together with Ohm's law enables the analysis of complex circuits
  • KCL provides equations based on currents at nodes
  • KVL provides equations based on voltages around loops
• Kirchhoff's laws are fundamental to circuit analysis and hold for any circuit configuration

Power in DC Circuits

• Power dissipated by a resistor equals the product of voltage across and current through the resistor: $P = IV$
  • Resistors convert electrical energy into heat energy
• Power can also be expressed in terms of resistance and either current or voltage: $P = I^2R = \frac{V^2}{R}$
• Total power in a circuit equals the sum of powers dissipated by individual components: $P_{total} = P_1 + P_2 + \ldots$
  • Conservation of energy implies that the total power supplied by sources equals the total power dissipated by components
• Efficiency $\eta$ is the ratio of useful output power to total input power: $\eta = \frac{P_{out}}{P_{in}}$
  • Ideal circuits have an efficiency of 100% with no power lost to heat
  • Real circuits always have some power loss due to resistance in wires and components
• Maximum power transfer theorem states that a source delivers maximum power to a load when the load resistance equals the source resistance

Circuit Analysis Techniques

• Equivalent resistance simplifies series-parallel circuits by combining resistors in series and parallel
  • Resistors in series add: $R_{eq} = R_1 + R_2 + \ldots$
  • Resistors in parallel add reciprocals: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots$
• Voltage division determines the voltage across a component in a series circuit: $V_x = \frac{R_x}{R_{total}}V_{total}$
• Current division determines the current through a component in a parallel circuit: $I_x = \frac{R_{eq}}{R_x}I_{total}$
• Nodal analysis applies KCL to find voltages at nodes using reference node (ground)
  • Write KCL equations for each node in terms of node voltages and component resistances
  • Solve system of equations to determine node voltages
• Mesh analysis applies KVL to find currents in loops using mesh currents
  • Define mesh currents for each loop and write KVL equations in terms of mesh currents and component resistances
  • Solve system of equations to determine mesh currents
• Superposition principle allows analyzing circuits with multiple sources by considering each source independently and summing the results
  • Deactivate all sources except one, analyze the circuit, and repeat for each source
  • Sum the individual contributions to find the total response

Practical Applications and Examples

• Electrical grids distribute power from generators to consumers using transformers to step up voltage for transmission and step down for distribution
  • High voltage transmission minimizes power loss due to resistance in power lines
• Batteries convert chemical energy into electrical energy to power devices (smartphones, laptops)
  • Rechargeable batteries can be recharged by applying a voltage to reverse the chemical reaction
• Sensors (thermistors, photoresistors) change resistance based on environmental conditions enabling measurement and control
  • Thermistors decrease resistance as temperature increases allowing temperature sensing
  • Photoresistors decrease resistance as light intensity increases enabling light detection
• Potentiometers are adjustable voltage dividers used for volume control, dimming, and sensor calibration
  • Adjusting the potentiometer changes the output voltage in proportion to the input voltage
• Fuses and circuit breakers protect circuits from excessive current that could cause damage or fire
  • Fuses contain a wire that melts and breaks the circuit when current exceeds a threshold
  • Circuit breakers use an electromagnet to trip a switch when current exceeds a threshold
• Kirchhoff's laws are used in circuit design to ensure proper operation and prevent component damage
  • KCL ensures currents are balanced at each node preventing charge accumulation
  • KVL ensures voltages are consistent around each loop preventing over-voltage conditions