Principles of Physics II

🎢Principles of Physics II Unit 5 – DC Circuits

DC circuits form the foundation of electrical engineering, enabling the flow and control of electric current. This unit explores key concepts like voltage, current, and resistance, as well as circuit components and analysis techniques. Students learn to apply Ohm's Law, Kirchhoff's Laws, and various circuit analysis methods to solve complex circuit problems. The unit also covers practical applications, from power distribution grids to everyday electronic devices, bridging theory and real-world implementation.

Key Concepts and Definitions

  • Electric current II represents the flow of electric charge through a conductor measured in amperes (A)
  • Voltage VV 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 RR quantifies the opposition to current flow in a conductor measured in ohms (Ω\Omega)
    • Conductors have low resistance allowing current to flow easily while insulators have high resistance impeding current flow
  • Power PP 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 VV equals current II times resistance RR: V=IRV = 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 ρ\rho, length LL, and cross-sectional area AA: R=ρLAR = \frac{\rho L}{A}
    • Longer wires have higher resistance while thicker wires have lower resistance for the same material
  • Conductivity σ\sigma is the reciprocal of resistivity: σ=1ρ\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: Vtotal=V1+V2+V_{total} = V_1 + V_2 + \ldots
    • Equivalent resistance equals the sum of individual resistances: Req=R1+R2+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: Itotal=I1+I2+I_{total} = I_1 + I_2 + \ldots
    • Reciprocal of equivalent resistance equals the sum of reciprocals of individual resistances: 1Req=1R1+1R2+\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: Iin=Iout\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: V=0\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=IVP = IV
    • Resistors convert electrical energy into heat energy
  • Power can also be expressed in terms of resistance and either current or voltage: P=I2R=V2RP = I^2R = \frac{V^2}{R}
  • Total power in a circuit equals the sum of powers dissipated by individual components: Ptotal=P1+P2+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 η\eta is the ratio of useful output power to total input power: η=PoutPin\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: Req=R1+R2+R_{eq} = R_1 + R_2 + \ldots
    • Resistors in parallel add reciprocals: 1Req=1R1+1R2+\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: Vx=RxRtotalVtotalV_x = \frac{R_x}{R_{total}}V_{total}
  • Current division determines the current through a component in a parallel circuit: Ix=ReqRxItotalI_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


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AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
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