Circuits


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Current


Conductivity
  • Unit of conductivity is the in Siemens/m, while Conductance is the reciprocal of resistance and has unit of Siemens.
  • Metallic conductivity: seen in solid metals or molten metals with some salts. Characterized by the metallic bond, which can be visualized as a sea of electrons flowing over a lattice of metal cations.
    • Good electrical and thermal conductors
    • Metal atoms can easily lose one or more of their outer electrons
  • Electrolytic Conductivity: This depends on the strength of the solution, but is otherwise similar to metallic conductivity.
    • Conductivity can be measured by placing the solution as a resistor and measuring the change in voltage across the solution. Can also be used to determine the ionic concentrations of solution since it is directly related to conductivity.

Current
  • The current is considered the flow of positive charge, even though only negative charges move. Has unit of ampere [1 A = 1 C/s]
    • I = Q/(Δt)
  • Electrons move from a point of lower electric potential to that of a higher electric potential. Thus the direction of the current is opposite to that of the electron flow.
  • Direct Current (DC): charge flows in one direction only
  • Alternating Current (AC): flow changes direction periodically.
  • Potential Difference (Voltage): can be produced by an electrical generator (galvanic cells)
    • Voltage is called the electromotive force in units of J/C.

Circuit Laws
  • Charge and energy must be fully accounted for at all times and can be neither created nor destroyed.
  • Kirchhoff’s Junction Rule: At any junction in a circuit, the sum of currents directed into that point equal the sum of currents leaving that point.
  • Kirchhoff’s Loop Rule: Around any closed loop, the sum of voltage sources will always be equal to the sum of voltage drops.

Resistance

  • The opposition within any material to the movement and flow of charge. Conductive materials that off a moderate amount of controllable resistance are called resistors.

Properties of Resistors
  • Depends on resistivity, length, cross-sectional area and temperature: R = (ρL)/A
  • Resistivity: Characterizes the intrinsic resistance of materials [Wm]
  • Length: a longer resistor means that the electrons will have to travel a greater distance. If length doubles, resistance doubles.
  • Cross-Sectional Area: Increases the number of conduction pathways through the resistor, which subsequently reduces resistance.
  • Temperature: Most conductors have greater resistance at higher temperatures. Higher temperature increases the amount of thermal vibration which leads to greater resistance to electron flow.

Ohms Law and Power
  • Ohm’s Law: For a given resistance, the voltage drop across a resistor will be proportional to the magnitude of the current.
    • V = IR
  • Even emf sources have intrinsic resistance, so the actual voltage of a battery can be calculated from:
    • V = Ecell – irint
  • Measuring Power: The following equation provides that rate at which energy is dissipated by a resistor:
    • P = IV = I^2R = V^2/R

Resistors in Series and Parallel
  • Resistors in Series: Current travels through each resistor in order to return to the cell.
    • Voltage Drop across cell will be: Vs = V1 + V2 + V3 + … + Vn
    • Resistance in total: Rs = R1 + R2 + R3 + … + Rn
  • Resistors in Parallel: Electrons have a choice regarding which path they take. The resistors are all wired with a common high-potential terminal and a common low potential terminal. Since they share these common terminals the voltages are the same across each division of resistors.
    • This functions to reduce the overall resistance by providing a greater number of conduction paths.
    • RP will always decrease as more resistors are added.
    • When n identical resistors are wired in parallel, the equivalent resistance becomes R/n.
      • 1/Rp = 1/R1 + 1/ R2 + … + 1/Rn

Capacitance and Capacitors

  • Capacitors have the ability to hold charge at a particular voltage. Will focus on parallel plate capacitor. Functions to store an amount of energy in the form of charge.

Properties of Capacitors
  • Positive charges buildup to plate connected to the positive terminal, and negative charges buildup on plate connected to negative terminal. These plates can store a charge at a particular voltage.
  • Capacitance of a capacitor is the ratio of the magnitude of the charge stored on one plate to the potential difference across the capacitor.
    • C = Q/v
    • Has units of the farad [1 F = 1 C/V]. Usually given in microfarads (10-6) or picofarads (10-12).
    • Parallel plates: C = εo(A/d): εo is the permittivity of free space (8.85 x 10-12 F/m)
      • Separation of charge sets up a uniform electric field: E = V/d with a directional arrow pointing from the positive plate to the negative plate.
  • Potential Energy:
    • U = 1/2CV^2

Dielectric Materials
  • Just another way of saying insulation. When a dielectric material is placed between the plates of a capacitor, it increases the capacitance by a factor called the dielectric constant (k).
  • A dielectric material will never decrease the capacitance.
  • Dielectrics in Isolated Capacitors: Voltage across the capacitor decreases when dielectric material is placed between the plates since the material shields the opposite charges from each other.
    • Increases capacitance through a decreased in voltage
  • Dielectrics in Circuit Capacitors: Charge on capacitor increases when material is placed within it. Voltage must remain constant, so instead the charge increases.

Capacitors in Series and Parallel
  • Series: Total capacitance decreases since the capacitors must share the voltage drop in the loop and therefore cannot store as much charge. These acts like an equivalent capacitor with a much larger distance between the plates.
    • 1/Cs = 1/C1 + 1/C2 + … + 1/Cn
  • Parallel: Voltage across each parallel capacitor is the same and this increase the equivalent capacitance.
    • Cp = C1 + C2 + … + Cn

Meters

  • Ammeters: used to measure the current at some point within a circuit. These require the circuit to be on, and must be put in series with the current that is to be measured.
    • Ideal ammeters will have zero resistance and no voltage drop across them
  • Voltmeters: Requires circuit to be active, uses magnetic properties of current carrying wires. Measures voltage drop across two points in a circuit and they must be wired in parallel to these two points.
    • An ideal voltmeter would have infinite resistance.
  • Ohmmeters: Do not require a circuit to be active. Usually have own battery with a known voltage, the ohmmeter then acts like an ammeter and uses Ohm’s law to find the resistance since the voltage is already known.