Unit 11: Electric Circuits

1. Electric Current and Current Density

Electric current is the flow of electric charge through a conductor, while current density describes the current flow per unit area.

I = dQ/dt
J = I/A = nqvd
where:
I = current (amperes)
J = current density (A/m²)
n = charge carrier density
vd = drift velocity
Important Concepts:
  • Conventional current flows from positive to negative
  • Electron flow is opposite to conventional current
  • Current is conserved at junctions (Kirchhoff's Current Law)
  • Drift velocity is typically very small

2. Resistance and Ohm's Law

Resistance describes how a material opposes the flow of electric current, and Ohm's Law relates voltage, current, and resistance.

V = IR
R = ρL/A
where:
R = resistance (ohms, Ω)
ρ = resistivity
L = length
A = cross-sectional area
Example: A copper wire (ρ = 1.68×10⁻⁸ Ω·m) of length 2m and diameter 0.5mm carries a current of 1.5A. Calculate:
a) Resistance
b) Voltage drop across wire

3. Electric Power and Energy

Electric power is the rate at which energy is transferred in a circuit.

P = IV = I²R = V²/R
Energy = Pt
Power Concepts:
  • Power is measured in watts (W)
  • Energy is measured in joules (J) or kilowatt-hours (kWh)
  • Power is dissipated as heat in resistors
  • Maximum power transfer occurs when load resistance equals source resistance

4. Kirchhoff's Laws

Kirchhoff's Laws are fundamental principles for analyzing complex circuits.

KCL: ΣI_in = ΣI_out
KVL: ΣV = 0 (around any closed loop)
Problem-Solving Steps:
  1. Label currents and voltage drops
  2. Apply KCL at junctions
  3. Apply KVL around loops
  4. Solve system of equations
Kirchhoff's Laws example circuit

5. Series and Parallel Circuits

Understanding how components combine in series and parallel is crucial for circuit analysis.

Series Resistance: Rₑq = R₁ + R₂ + R₃ + ...
Parallel Resistance: 1/Rₑq = 1/R₁ + 1/R₂ + 1/R₃ + ...
Example: Calculate the equivalent resistance of:
a) Three 6Ω resistors in series
b) Three 6Ω resistors in parallel

6. RC Circuits

RC circuits contain both resistors and capacitors, exhibiting time-dependent behavior.

Charging: V(t) = V₀(1 - e^(-t/RC))
Discharging: V(t) = V₀e^(-t/RC)
Time constant: τ = RC
RC Circuit Properties:
  • Capacitor voltage cannot change instantaneously
  • Current changes instantly when switch is closed/opened
  • After 5τ, circuit is essentially at steady state
  • Energy is stored in capacitor's electric field

7. Ammeters and Voltmeters

Understanding how to properly use measuring devices is essential for circuit analysis.

Measurement Guidelines:
  • Ammeters connected in series with circuit element
  • Voltmeters connected in parallel across circuit element
  • Ideal ammeter has zero resistance
  • Ideal voltmeter has infinite resistance
  • Real meters affect circuit operation

8. Complex Circuit Analysis

Advanced techniques for analyzing more complicated circuits.

Analysis Methods:
  • Mesh analysis
  • Node voltage analysis
  • Superposition principle
  • Thévenin's theorem
  • Norton's theorem
Example: Use Thévenin's theorem to find the current through a 5Ω resistor connected across points A and B of a complex network.

9. Practical Applications

Understanding how circuit principles apply to real-world devices and systems.

Common Applications:
  • Voltage dividers
  • Current dividers
  • Filters and signal processing
  • Power supplies
  • Battery charging circuits
  • LED circuits

10. Practice Problems and Review

Problem 1: A circuit contains a 12V battery and three resistors (4Ω, 6Ω, 12Ω) in parallel. Calculate:
a) Equivalent resistance
b) Total current
c) Power dissipated in each resistor
Problem 2: In an RC circuit with R = 100kΩ and C = 47μF:
a) Find the time constant
b) Calculate voltage after one time constant if V₀ = 9V
c) Determine time to reach 90% of final voltage
Review Strategy:
  • Master fundamental concepts first
  • Practice both qualitative and quantitative problems
  • Draw clear circuit diagrams
  • Check answers using multiple methods
  • Verify units and reasonable values