Chapter 4 - Electric Circuits

Current

• Current: the amount of charge passing a point in a given time period
  • ==I = ΔQ/Δt==
  • I: current (Amperes)
  • Q: charge (Coulombs)
  • t: time (seconds)
  • Current is described in the AP exam as the flow of positive charge

Ohm’s Law

• Batteries: create currents using a difference in potential
  • The “+” terminal has a higher electric potential
  • The “-” terminal has a lower electric potential
  • Generally, the greater the potential difference, the more current flows
• Resistance: a property of the circuit that resists the current
  • Units are Ohms (Ω)
  • ==R = ⍴L/A==
  • R: resistance
  • ⍴: resistivity
  • L: length of wire
  • A: cross-sectional area of wire
• Ohm’s Law
  • ==I = ΔV/R==
  • ΔV: voltage across a certain part of the circuit (like a resistor)
  • R: resistance
  • I: current
• Power: the rate at which electrical energy is converted to heat energy
  • ==P = IΔV = I^2 R = ΔV^2/R==
• Ohmic vs. Nonohmic
  • Ohmic: a circuit part (resistor or capacitor) that maintains the same resistance when the voltage across it or current through it changes - resistance is constant
• Circuit Pictures
  • Wire: straight line
  • Battery: 4 parallel lines - one long line and then one smaller line repeated
  • Resistor: zig zag line
  • Capacitor: 2 parallel lines
• Resistors in Series
  • ==R = ⅀Ri==
  • Ri: the resistances of the resistors in series with each other
  • R: equivalent resistance or total resistance
• Resistors in Parallel
  • ==1/R = ⅀1/Ri==
  • Ri: the resistances of the resistors in parallel with each other
  • R: equivalent resistance or total resistance
• Rules for resistors in circuits
  • The current in resistors in series is equal to each other
  • The voltage across resistors in parallel is equal to each other
  • The voltage across two resistors in parallel is also equal to the voltage across each individual resistor
• V-I-R charts
  • Create columns of V, I, and R for each individual resistor and the total circuit
  • This helps us stay organized in complex problems

Kirchhoff’s Rules

• Junction Rule: The current entering and leaving a junction is equal
• Loop Rule: In a closed loop, the sum of the voltages is 0
  • Choose a loop of the circuit and when you see a resistor, the voltage is -IR because resistors resist the current
  • If the loop is against the current, the voltage is +IR
  • When you see a batter, add the voltage of the battery (if going from - to +)
  • If you go from + to -, subtract the voltage of the battery
  • If the current you calculate is negative, you chose the wrong direction and the current flows the opposite way

Experimental Circuits

• In calculations, we assume most electronic devices in circuits act as resistors
• Light bulb
  • The brightness of a bulb depends only on the power dissipated
  • The power of a bulb can change depending on the current and voltage of the circuit it’s in
• Ammeters and Voltmeters
  • Ammeters: measure current
  • Ammeters work by putting them in series with resistors (current is constant for resistors in series)
  • Voltmeters: measure potential difference (voltage)
  • Voltmeters work by putting them parallel to parallel resistors
• Real batteries
  • In a perfect world, batteries have no resistance but in the real world, this is not true
  • The voltage advertised by a battery, ε, is actually larger than the real voltage ΔV (terminal voltage)
  • ==ΔV = ε - Ir==
    • r: the internal resistance of the battery
    • I: current through the battery
  • Internal resistance is measured by hooking a battery up to a resistor and plotting the terminal voltage of the battery as a function of the current through the battery
    • The slope will be equal to -r
• Switches
  • Open switch: that part (loop) of the circuit can be considered gone (dead)
• Capacitors
  • Capacitors: two parallel metal plates separated by either air or dielectric material
  • Capacitance: how much charge a capacitor can hold for each volt of potential difference
  • ==C = kεA/d==
    • C: capacitance
    • k: dielectric constant
    • ε: vacuum permittivity constant
    • A: area of one of the plates (both plates have the same area)
    • d: distance between plates
  • ==ΔV = Q/C==
  • ΔV: Voltage
  • Q: charge
  • C: capacitance (Farads)
  • ==U = 1/2 QΔV = 1/2 C(ΔV )^2==
  • U: energy stored in a capacitor
  • Q: charge
  • ΔV: potential difference
  • C: capacitance
  • ==ΔV/Δr = E==
  • E: electric field
  • ΔV: potential difference
  • Δr: distance between plates
• Parallel vs. Series Capacitors
  • Parallel Capacitors
  • ==C = ⅀Ci==
    • C: total capacitance for capacitors in parallel
    • Ci: capacitance of capacitors in parallel
    • This is the same formula from resistors in series - capacitors are basically resistors in reverse
  • Series Capacitors
  • ==1/C = ⅀1/Ci==
    • C: total capacitance for capacitors in series
    • Ci: capacitance of capacitors in series

RC Circuits

• RC Circuit: a circuit containing resistor(s) and capacitor(s)
• You’ll only be asked about RC Circuits in certain states
  • When you first connect a capacitor to a circuit:
  • No charge has built up so treat the capacitor like a wire with no potential difference
  • After a long time:
  • The capacitor has charged up to its max so no current will flow through it
  • Treat the capacitor like an open switch
  • The potential difference across the capacitor equals the voltage of the devices parallel to the capacitor