⚡️College Physics III – Thermodynamics, Electricity, and Magnetism Unit 10 – Direct-Current Circuits

Key Concepts and Definitions

• Electric current ($I$) is the flow of electric charge through a conductor measured in amperes (A)
• Voltage ($V$) is the potential difference between two points in an electric circuit measured in volts (V)
  • Voltage provides the driving force for electric current to flow
• Resistance ($R$) is the opposition to the flow of electric current in a conductor measured in ohms ($\Omega$)
  • Conductors with low resistance allow current to flow easily (copper wire) while insulators have high resistance and restrict current flow (rubber)
• Power ($P$) is the rate at which electrical energy is converted into other forms of energy (heat, light, mechanical) measured in watts (W)
• Direct current (DC) circuits involve the unidirectional flow of electric current from a source (battery) to a load (resistor)
• Alternating current (AC) circuits involve the periodic reversal of current direction and are used in household electrical systems
• Kirchhoff's laws describe the behavior of current and voltage in electric circuits
• Ohm's law relates voltage, current, and resistance in a linear circuit component

Circuit Components and Symbols

• Voltage source (battery) provides the potential difference to drive current flow represented by two parallel lines with + and - labels
• Resistor opposes current flow and dissipates electrical energy as heat represented by a zigzag line
• Capacitor stores electric charge and energy in an electric field represented by two parallel lines
• Inductor stores energy in a magnetic field and opposes changes in current represented by a coil or spiral
• Switch controls the flow of current by opening or closing a circuit represented by a line with a break
• Ammeter measures electric current and is connected in series with the circuit component represented by a circle with an A inside
• Voltmeter measures voltage and is connected in parallel with the circuit component represented by a circle with a V inside
• Wires are conductors that connect circuit components and allow current to flow represented by solid lines

Ohm's Law and Resistance

• Ohm's law states that the voltage ($V$) across a resistor is directly proportional to the current ($I$) flowing through it: $V = IR$
  • Doubling the voltage across a resistor will double the current flowing through it if the resistance remains constant
• Resistance is a material property that depends on the resistivity ($\rho$), length ($L$), and cross-sectional area ($A$) of the conductor: $R = \rho L/A$
  • Increasing the length of a wire increases its resistance while increasing its cross-sectional area decreases its resistance
• Resistors can be connected in series (end-to-end) or parallel (side-by-side) to create equivalent resistances
  • Series resistors have the same current flowing through them and their equivalent resistance is the sum of individual resistances: $R_{eq} = R_1 + R_2 + ...$
  • Parallel resistors have the same voltage across them and their equivalent resistance is the reciprocal of the sum of reciprocals: $1/R_{eq} = 1/R_1 + 1/R_2 + ...$
• Temperature affects the resistance of materials with most metals increasing in resistance as temperature rises (positive temperature coefficient) while some materials like carbon decrease in resistance with increasing temperature (negative temperature coefficient)

Series and Parallel Circuits

• Series circuits have components connected end-to-end forming a single path for current to flow
  • Current is the same through all components in a series circuit
  • Voltage drops across each component and the sum of voltage drops equals the source voltage: $V_{source} = V_1 + V_2 + ...$
  • Equivalent resistance of series components is the sum of individual resistances: $R_{eq} = R_1 + R_2 + ...$
• Parallel circuits have components connected side-by-side forming multiple paths for current to flow
  • Voltage is the same across all components in a parallel circuit
  • Current divides among the parallel branches according to Ohm's law: $I_{total} = I_1 + I_2 + ...$
  • Equivalent resistance of parallel components is the reciprocal of the sum of reciprocals: $1/R_{eq} = 1/R_1 + 1/R_2 + ...$
• Combination circuits contain both series and parallel connections and can be analyzed by simplifying the circuit into equivalent resistances
  • Identify series and parallel sections, calculate their equivalent resistances, and redraw the circuit with the simplified components
  • Repeat the process until the circuit is reduced to a single equivalent resistance

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: $\Sigma I_{in} = \Sigma I_{out}$
  • A node is a point where two or more circuit elements connect
  • KCL is based on the conservation of electric charge, which cannot accumulate at a node
• Kirchhoff's voltage law (KVL) states that the sum of voltage drops around any closed loop in a circuit equals zero: $\Sigma V = 0$
  • A closed loop is any path that starts and ends at the same node
  • KVL is based on the conservation of energy, as the net work done around a closed loop must be zero
• Kirchhoff's laws are used to analyze complex circuits by setting up and solving systems of linear equations
  • Assign variables to unknown currents and voltages, apply KCL at nodes and KVL around loops, and solve the resulting equations
• Kirchhoff's laws are consistent with Ohm's law and the principles of series and parallel circuits
  • Applying KCL at a node in a parallel circuit yields the current divider formula: $I_{total} = I_1 + I_2 + ...$
  • Applying KVL around a loop in a series circuit yields the voltage divider formula: $V_{source} = V_1 + V_2 + ...$

Power and Energy in DC Circuits

• Electric power ($P$) is the rate at which electrical energy is converted into other forms of energy (heat, light, mechanical) measured in watts (W)
  • Power is the product of voltage and current: $P = VI$
  • Power can also be expressed in terms of resistance using Ohm's law: $P = I^2R = V^2/R$
• Energy ($E$) is the capacity to do work and is measured in joules (J) or watt-seconds (Ws)
  • Electrical energy is the product of power and time: $E = Pt$
  • The cost of electrical energy is typically measured in kilowatt-hours (kWh), where 1 kWh = 3.6 MJ
• Power dissipation in resistors occurs when electrical energy is converted into heat due to the collision of electrons with atoms in the material
  • The power dissipated by a resistor is proportional to the square of the current flowing through it: $P = I^2R$
  • Resistors have power ratings that specify the maximum power they can dissipate without overheating or failing
• Efficiency ($\eta$) is the ratio of useful output power to total input power expressed as a percentage: $\eta = P_{out}/P_{in} \times 100\%$
  • Efficient devices (LED lights) convert a large fraction of input power into useful output power while inefficient devices (incandescent bulbs) waste a significant portion as heat
• Maximum power transfer occurs when the load resistance equals the source resistance in a circuit
  • Adjusting the load resistance to match the source resistance maximizes the power delivered to the load but results in a 50% efficiency

Circuit Analysis Techniques

• Equivalent resistance simplifies complex circuits by combining series and parallel resistors into a single equivalent resistance
  • For series resistors: $R_{eq} = R_1 + R_2 + ...$
  • For parallel resistors: $1/R_{eq} = 1/R_1 + 1/R_2 + ...$
• Voltage divider circuits consist of two or more resistors in series and are used to produce a fraction of the input voltage
  • The output voltage is proportional to the ratio of the target resistor to the total resistance: $V_{out} = V_{in} \times (R_{target}/R_{total})$
  • Voltage dividers are used in sensors (thermistors, photoresistors) and to provide reference voltages in electronic circuits
• Current divider circuits consist of two or more resistors in parallel and are used to split the input current into fractional output currents
  • The output current through each branch is inversely proportional to its resistance: $I_{branch} = I_{total} \times (R_{total}/R_{branch})$
  • Current dividers are used in current mirrors and to balance the load on power supplies
• Mesh analysis is a circuit analysis technique that applies Kirchhoff's voltage law (KVL) to loops in a circuit
  • Assign a mesh current to each loop, apply KVL to express voltages in terms of mesh currents, and solve the resulting equations
• Nodal analysis is a circuit analysis technique that applies Kirchhoff's current law (KCL) to nodes in a circuit
  • Assign a node voltage to each non-reference node, apply KCL to express currents in terms of node voltages, and solve the resulting equations
• Superposition principle states that the response of a linear circuit to multiple sources is the sum of its responses to each source individually
  • To analyze a circuit with multiple sources using superposition:
    1. Turn off all sources except one
    2. Analyze the circuit to find the response (voltage or current) due to the active source
    3. Repeat steps 1 and 2 for each source
    4. Sum the individual responses to find the total response

Real-World Applications

• Household wiring uses parallel circuits to distribute electricity to multiple devices while maintaining a constant voltage (120 V or 240 V)
  • Outlets, switches, and light fixtures are wired in parallel to allow independent control and to prevent a single device failure from affecting others
  • Circuit breakers and fuses protect against overloads and short circuits by interrupting the current flow when it exceeds a safe level
• Batteries are series or parallel combinations of electrochemical cells that convert chemical energy into electrical energy
  • Series connections (flashlight batteries) increase the voltage while maintaining the capacity
  • Parallel connections increase the capacity while maintaining the voltage
  • Battery management systems (in electric vehicles) monitor and balance the charge of individual cells to ensure safe and efficient operation
• Solar panels are series and parallel combinations of photovoltaic cells that convert light energy into electrical energy
  • Series connections (strings) increase the voltage to match the inverter input requirements
  • Parallel connections (arrays) increase the current and power output
  • Maximum power point tracking (MPPT) optimizes the load resistance to extract the maximum power from the solar panels under varying light conditions
• Sensors use the principles of resistance, voltage division, and current division to convert physical quantities (temperature, light, pressure) into electrical signals
  • Thermistors are temperature-sensitive resistors that change resistance with temperature, allowing temperature measurement using a voltage divider circuit
  • Photoresistors (light-dependent resistors) change resistance with light intensity, enabling light sensing applications
  • Strain gauges are resistors that change resistance when stretched or compressed, used in pressure sensors and load cells
• Potentiometers are adjustable voltage dividers used to control the output of a circuit
  • The wiper (moving contact) position determines the fraction of the input voltage that appears at the output
  • Potentiometers are used as volume controls, dimmer switches, and in analog control systems
• Kirchhoff's laws and circuit analysis techniques are used to design and troubleshoot complex electronic systems
  • Analog circuits (amplifiers, filters, power supplies) are analyzed using Kirchhoff's laws, Ohm's law, and circuit theorems
  • Digital circuits (logic gates, microprocessors) are analyzed using Boolean algebra and Kirchhoff's laws to ensure proper operation and to minimize power consumption