Unit C-10 – Circuits and Capacitors Lecture Notes

Ohm’s Law

  • Definition of Current:

    • Current is the flow of electric charge.

    • Symbol: I

    • Units: Amperes (A)

    • Relationship: I = Q/t (where Q is charge in coulombs and t is time in seconds)

  • Resistance:

    • Measured in Ohms (Ω).

    • Ohm's Law: V = I × R (V = Voltage, I = Current, R = Resistance)

    • Power delivered:

      • P = I × V

      • P = I² × R

      • P = V² / R

Kirchhoff’s Rules

  • Junction Rule:

    • At any junction, the total current entering is equal to the total current leaving. (Based on the Law of Charge Conservation)

  • Series Rule:

    • Total resistance for series resistors: Rᵢ = R₁ + R₂ + … + Rₙ

  • Loop Rule:

    • The sum of all voltage drops around a closed loop must equal the sum of all voltage sources. (Based on the Law of Energy Conservation)

  • Parallel Rule:

    • Total current for parallel resistors: Iᵢ = I₁ + I₂ + … + Iₙ

Combining Resistors

Series Resistors

  • Total Resistance:

    • Rₛ = R₁ + R₂ + … + Rₙ

    • Voltage Drop: V = V₁ + V₂ + … + Vₙ

Parallel Resistors

  • Total Resistance:

    • 1/Rₚ = 1/R₁ + 1/R₂ + … + 1/Rₙ

    • Current: I = I₁ + I₂ + … + Iₙ

Voltmeters and Ammeters

  • Ammeter:

    • Measures current.

    • Internal resistance: ideally 0.

    • Connected in series with the component.

  • Voltmeter:

    • Measures voltage.

    • Internal resistance: ideally infinity.

    • Connected in parallel with the element.

  • Example Calculations:

    1. With ideal meters: Calculate current, voltage, power across components.

    2. With non-ideal meters: Adjust calculations according to the internal resistances of the meters.

Drift Velocity of Charge Carriers

  • Charge Carriers: Particles that carry charge, such as electrons.

  • Drift Velocity (vₖ):

    • Average speed of charge carriers in the direction of the electric current.

    • Relationship between current (I), charge density (n), and A (cross-sectional area): I = n × A × vₖ.

    • Calculate average drift velocity using provided charge density data.

Current Density and Electric Field

  • Current Density (J):

    • Measured as current per unit area: J = I / A

    • Units: Amperes per square meter (A/m²).

  • Electric Field (E):

    • In a conductor: E = ρ × J where ρ is resistivity.

Capacitance – Storing Electrical Energy

  • Capacitor: Stores electrical energy through separation of charges.

    • Capacitance (C): C = Q / V (where Q is charge and V is voltage).

    • Units: Farads (F).

    • Dependence: Geometric factors of capacitor (shape, spacing).

  • Deriving Capacitance for Specific Configurations (Parallel Plate, Spherical, and Cylindrical).

  • Energy Stored: U = 1/2 × C × V².

RC Circuits

  • Rate of Charging/Discharging:

    • The time constant: τ = R × C (where R is resistance and C is capacitance).

    • Formulas for charge (Q), voltage (V), and current (I) over time.

  • Initial and Steady-State Behavior:

    • Immediately after capacitor charging, it acts like a short circuit, fully charged it acts like an open circuit.