Electromagnetic Induction Notes

Electromagnetic Induction Notes

Introduction to Flux

  • Electric Flux Analogy: Visualize holding a hoop in the rain, measuring the amount of rain passing through it.
    • Factors affecting electric flux:
    • Intensity of rain
    • Area of the hoop
    • Orientation of the hoop

Magnetic Flux

  • Definition: Magnetic flux is akin to electric flux but measures the magnetic field passing through a surface.
  • Calculation:
    • General formula: \Phi_B = \int \mathbf{B} \, \cdot \, d\mathbf{A}
    • For uniform magnetic field: \Phi_B = B A \cos(\theta)
    • Unit: Weber (Wb)
    • 1 Wb = 1 T·m²

Faraday’s Law of Electromagnetic Induction

  • Faraday's Discovery (1831): EMF (electromotive force) arises when magnetic flux changes through a loop.
  • Mathematical Representation: \mathcal{E} = -\frac{d\Phi_B}{dt}
    • Units: Wb/s = Volts
  • Example: If a loop's magnetic flux drops from 1.0 Wb to 0 Wb in 10 s, the induced EMF is:
    • \mathcal{E} = -\frac{(0 - 1.0 \, \text{Wb})}{10 \, \text{s}} = 0.1 \, \text{V}

Lenz’s Law

  • Definition: States that the direction of induced current opposes the change in magnetic flux that produced it.
    • Formula: \mathcal{E} = -\frac{d\Phi_B}{dt}
  • Current Direction: When a loop exits a magnetic field, Lenz’s Law predicts clockwise current as it opposes the change.

Induction in a Coil

  • Coil Representation: A coil acts as multiple loops in series.
  • Formula: \mathcal{E} = -N \frac{d\Phi_B}{dt}
    • Example Calculation: For a coil with 50 turns, in a magnetic field transitioning from 2 T to 6 T in 0.2 s, the induced EMF is found to be 17.7 V (current direction clockwise).

Simple Generators

  • Generator Mechanism: Spinning a loop in a magnetic field changes flux due to angle changes, creating an induced EMF regardless of how flux changes.
  • Combined Equation:
    • \mathcal{E} = -N B A \frac{d(cos(\theta))}{dt}
  • Example: A 10 cm² coil with 200 turns spinning in a 0.477 T magnetic field generates a maximum EMF of about 10 V.

Bar on Rails Model

  • Setup: Bar of length \ell sliding on conducting rails with a resistor completing the circuit.
  • Induced EMF: As the bar slides, the flux increases, resulting in a counterclockwise current.
  • Calculation:
    • \mathcal{E} = -B\ell\frac{dx}{dt}
    • For a 30 cm bar in a 2 T field moved 20 cm in 0.1 s with R = 2 Ω:
    • Induced EMF: 1.2 V
    • Current: 0.6 A
    • Power: 0.72 W
    • Work done: 0.072 J

Inductors

  • Definition: A passive device that opposes changes in current, creating an EMF when current changes.
  • Analogy: Compared to a water wheel that resists changes in water flow until it matches the flow speed.

LR Circuits

  • Components: Contains a battery, resistor, and inductor.
  • Current Growth: Can be described by the equations:
    • I(t) = \frac{\mathcal{E}}{R} \left(1 - e^{-\frac{Rt}{L}}\right)
    • \frac{dI}{dt} = \frac{\mathcal{E}}{L} e^{-\frac{Rt}{L}}
  • Time Constant: \tau = \frac{L}{R}
  • Example Calculation: For a 25 mH inductor with an 8.0 Ω resistor and a 6.0 V battery:
    • Time constant: 3.125 ms
    • Voltage drop across resistor at t=0: 0 V, after one time constant: 3.79 V.

LC Circuits

  • Charging and Discharging: When a capacitor discharges, charge oscillates due to inductor presence, creating energy oscillation between capacitor and inductor.
  • Stored Energy:
    • Capacitor: U_C = \frac{1}{2} C V^2
    • Inductor: U_L = \frac{1}{2} L I^2

Maxwell’s Equations

  • Foundation for electromagnetism containing the four key relations:
    1. Gauss’s Law for Electric Fields.
    2. Gauss’s Law for Magnetism.
    3. Maxwell-Faraday Equation (related to changing magnetic fields).
    4. Ampère-Maxwell Law (with displacement current).
  • Significance: Described the relationship between electricity and magnetism, predicting the existence of electromagnetic waves.