Electric Current and Circuits

Electric Current Basics

• An electric circuit includes a power source, a load (like a lamp), conducting wires, and a switch.
• Wires create an uninterrupted path for current to flow when the switch is closed.
• The battery is represented by parallel lines in a schematic, with longer lines indicating the positive terminal.
• A lamp is represented by a circle encompassing a filament.
• Closing the switch completes the circuit, allowing current to flow from the positive to the negative terminal of the battery.
• Conventional current is depicted as flowing from positive to negative.
• In metal wires, current is carried by electrons (negative charges), while in ionic solutions, both positive and negative charges move.
• The movement of charged particles flowing through a wire composes a current.

Benjamin Franklin's Contribution

• Benjamin Franklin theorized current flowed from a material with more "electrical fluid" (positive) to one with less (negative).
• He defined positive as having more fluid and negative as lacking it.
• Franklin surmised that current flowed from the positive to the negative material, which he termed a positive current flow.

Current, Electrical Fields, and Equilibrium

• Current (I) is the rate at which charge moves through an area (A) of a wire's cross-section and is defined to move in the direction of the electrical field.
• Conventional current is in the direction of the electrical field.
• Electron flow (electronic flow) goes in the opposite direction.
• A material is positive if it has more protons than electrons, and negative if it has more electrons than protons.
• In conducting metals, current flow is primarily due to electrons moving from negative to positive.
• Electrical fields exist in conductors and drive current.
• In static electricity, charges redistribute to cancel external electrical fields and restore equilibrium.
• In circuits, a battery or another electric potential source prevents charges from reaching equilibrium, thus maintaining current flow.

Electrical Circuits

• Electrical circuits are essential in many complex systems like skyscrapers, airplanes, and electronics.
• Modern circuits are often made of conducting material on insulating boards.
• Circuit boards operate on similar principles as simpler series circuits.

Parallel Circuits

• Charges in parallel circuits have multiple paths to return to the source.
• Current leaves the source (e.g., a battery), passes through a switch, and divides through multiple paths (e.g., light bulbs) before rejoining.
• Each electron passes through only one path in the circuit.
• In parallel circuits, the voltage drop across each resistor is equal to the total voltage drop: VT=V1=V2=V3.
• The total current is the sum of the individual currents through each resistor: IT=I1+I2+I3

Ohm's Law in Parallel Circuits

• Ohm's Law still applies: the current through each resistor equals the total voltage drop divided by the resistance of that resistor (I=V/R).
• The total resistance (aka equivalent resistance) for parallel circuits is given by: 1/RT=1/R1+1/R2+1/R3

Parallel Circuit Example

• A circuit has a 90V voltage drop, with three parallel resistors: 60 Ω, 30 Ω, and 30 Ω.
• Each resistor also has a 90V voltage drop.
• The currents through each resistor are:
  • I1=90V/60Ω=1.5A
  • I2=90V/30Ω=3.0A
  • I3=90V/30Ω=3.0A
• The total current is the sum of the individual currents: IT=1.5A+3.0A+3.0A=7.5A
• The equivalent resistance can be calculated by two methods.
  • 1/RT=1/60 Ω/5= 12 Ω
  • RT=VT/IT =90. Ω/7.5 A = 12 Ω

Conservation Laws

• Expressions for series and parallel resistance rely on the laws of conservation of energy and charge.
• These laws state that total charge and energy remain constant during any process.
• The total energy is conserved in a circuit and expressed as: 𝑞𝑉=𝑞𝑉1+𝑞𝑉2+𝑞𝑉3
• Simplified it becomes 𝑉=𝑉1+𝑉2+𝑉3
• Total amount of charge passes through both the battery and each resistor, and so there is no loss/leakage

Series Resistance

• In series, 𝑉=𝐼𝑅1+𝐼𝑅2+𝐼𝑅3 = 𝐼(𝑅1+𝑅2+𝑅3)
• Equivalent single series resistance: 𝑉=𝐼𝑅S
• Total or equivalent series resistance of three resistors: 𝑅S=𝑅1+𝑅2+𝑅3

Analyzing a Series Circuit

• A battery's voltage output is 12.0V, connected to series resistors: R1=1.00Ω, R2=6.00Ω, R3=13.0Ω
• Total resistance (Rs) is the sum of individual resistances: Rs=1.00Ω+6.00Ω+13.0Ω=20.0Ω
• Current (I) is voltage divided by total resistance: I=V/Rs =12.0V20.0Ω=0.600A
• Calculate the voltage drop in each resistor by finding the product of its Current and Resistance.
  • V1=IR1=(0.600A)(1.0Ω)=0.600V
  • V2=IR2=(0.600A)(6.0Ω)=3.60V
  • V3=IR3=(0.600A)(13.0Ω)=7.80V
• The sum of the voltage drops equals the source voltage: V1+V2+V3=(0.600+3.60+7.80)V=12.0V

Power Dissipation in Series Circuits

• Joule's law calculates power (P) in watts (W) dissipated by a resistor: P=IV (where power is electric power, and is the product of Current and Voltage).
• Each resistor has the same current.
• Power dissipated by the first resistor: - 𝑃1=𝐼^2𝑅1 - =(0.600A)^2(1.00Ω) - =0.360W
• The same formula applies to all other resistors as well.
• Power can be calculated by 𝑃=𝐼𝑉 or 𝑃=𝑉^2/𝑅 (where 𝑉 is the voltage drop across the resistor).
• Total power dissipated by the resistors is the same as the power put out by the source, and thus is simply the sum of power put out by each resistor: - 𝑃1+𝑃2+𝑃3 - =(0.360+2.16+4.68)W - =7.20W

Resistors in series

• Series resistances add: 𝑅S=𝑅1+𝑅2+𝑅3+…
• Same current flows through all resistors
• Individual resistors in a series do not get total source voltage, but divide it instead

Resistors in Parallel

• Each resistor is connected directly to the voltage source with negligible resistance in connecting wires.
• Each resistor gets full voltage
• Each resistor draws the same current had it been connected alone
• Automobile headlights, radio etc. are wired in parallel so that they retain full voltage
• Total current produced by the source is the sum of currents: 𝐼=𝐼1+𝐼2+𝐼3
• Ohm’s law for the equivalent single resistance gives
  • 𝐼=𝑉/𝑅p=𝑉(1/𝑅p)
  • Simplified: 1/𝑅p=1/𝑅1+1/𝑅2+1/𝑅3

Total Resistence in Parallel Equations

• Total resistance (𝑅p) is less than than the smallest of the individual resistances
• When resistors are connected in parallel, more current flows from the source than flows through resistors individually, which makes the total resistance lower.