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.