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