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

• 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)

• 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