Electromagnetic Induction

Lenz's law states: The direction of induced current is such that it opposes the change that induced it.

Key principles: • If flux is increasing: induced current creates a magnetic field that weakens the original flux • If flux is decreasing: induced current creates a magnetic field that strengthens the original flux

This law is a manifestation of energy conservation. The induced current always flows in a direction that creates a magnetic force opposing the motion/change, requiring work to be done against this opposition.

Example: When a magnet approaches a loop north-pole first, the induced current creates a magnetic field away from the magnet (effectively creating a north pole facing the approaching magnet).

Energy density in magnetic field: $u = \frac{B^2}{2\mu_0}$

Energy stored in an inductor: $U = \frac{1}{2}Li^2$

Relationship: • For a solenoid with inductance L = μ₀n²πr²l and field B = μ₀ni: • Total energy: $U = \frac{1}{2}Li^2 = \frac{1}{2}\mu_0n^2\pi r^2l \cdot i^2$ • This equals $\frac{B^2}{2\mu_0}V$ where V = πr²l (volume) • Therefore energy stored in inductor equals energy density × volume

Electromagnetic significance: • Energy is physically stored in the magnetic field itself, not in the conducting material • This field energy can be recovered (unlike resistive losses) • Represents potential energy in the electromagnetic field • Analogous to energy density $\frac{1}{2}\epsilon_0E^2$ in electric field • Forms basis for electromagnetic wave energy transport

This concept is fundamental to understanding transformers, motors, and electromagnetic radiation.

To determine the direction of induced current when a rectangular loop's area decreases in a magnetic field:

1. Identify the change: Decreasing area means decreasing magnetic flux through the loop
2. Apply Lenz's Law: The induced current must create a magnetic field that opposes this decrease
3. Since the original flux is decreasing, the induced current must produce a magnetic field in the same direction as the original field (into the plane)
4. Using the right-hand rule: When your thumb points in the direction of the induced magnetic field (into the plane), your fingers curl in the direction of the current

Therefore, the induced current will flow clockwise around the rectangular loop PQRS when viewed from the direction the magnetic field is pointing.

This creates a magnetic attraction that opposes the physical reduction of the loop's area.

A moving conductor in a magnetic field acts as an equivalent battery with:

• EMF = vBl (where v = velocity, B = magnetic field, l = conductor length)
• Internal resistance = resistance of the moving conductor itself
• Positive terminal at the end where positive charges accumulate
• Negative terminal at the end where electrons accumulate

Circuit behavior:

• The motional EMF drives current against resistance
• Current = vBl/(R + r) where R is external resistance and r is conductor resistance
• Power delivered equals power dissipated as heat (P = i²R)

This equivalence explains why mechanical energy must be continuously supplied to maintain the conductor's motion against magnetic forces, just as chemical energy is consumed in a battery to maintain potential difference.

• Motion-induced emf: E = vBl (where l is the length of the segment moving through the field)
• Current flow: i = E/R = vBl/R (clockwise direction as per Lenz's law)
• Magnetic force on moving wire: F₁ = ilB = (vBl/R)lB = vB²l²/R
• Direction: The magnetic force opposes the velocity (opposing the motion that produces it)
• Net effect: To maintain constant velocity, an external force F = F₁ = vB²l²/R must be applied
• Energy conservation: Power delivered by external force (F·v = v²B²l²/R) equals the thermal power dissipated in the loop (i²R = v²B²l²/R)

• The energy stored in an inductor is given by U = ½Li²
• As current follows i = i₀(1-e^(-t/τ)), the energy increases non-linearly
• Energy storage progression:
  • At t = 0: U = 0 (no energy stored)
  • At t = τ: U = ½L(0.63i₀)² ≈ 0.4Li₀²
  • At t = ∞: U = ½Li₀² (maximum energy)
• The rate of energy storage is highest at the beginning and gradually decreases
• During current growth, part of the battery's energy becomes stored in the inductor's magnetic field, while the rest is dissipated as heat in the resistor
• The inductor effectively delays energy transfer in the circuit, with τ controlling this delay's duration

Example: In a 20mH inductor with 10Ω resistance and 5A max current, after one time constant (2ms), approximately 0.4×(0.5×10^(-3)×25) = 5mJ is stored in the magnetic field

• The energy stored in an inductor is given by: $U = \frac{1}{2}Li^2$
• This energy is stored in the form of a magnetic field that builds up when current flows through the inductor
• When current changes in the inductor, this stored energy can be:
  • Released back into the circuit when current decreases
  • Increased when current increases
• The energy density in the magnetic field (energy per unit volume) is: $u = \frac{B^2}{2\mu_0}$
• Example: A 50 mH inductor carrying 2.0 A of current stores 0.10 J of energy

The growth of current in an LR circuit is governed by:

• Differential equation: $ε - L\frac{di}{dt} = Ri$

• Solution for current at time t: $i = \frac{ε}{R}(1 - e^{-t/τ})$

• Time constant τ = L/R represents:

  • The time needed for current to reach 63% of its maximum value
  • A measure of how quickly the circuit responds to changes
  • Larger L/R ratio means slower current growth

• Maximum current: i₀ = ε/R (reached theoretically at t = ∞)

• Physical significance: The time constant represents the competing effects between:

  • The inductor's opposition to current change
  • The resistor's opposition to current flow

The relationship between primary and secondary coils enables voltage amplification through:

• Coil arrangement: Secondary coil (S) is wound coaxially over the primary coil (T), both surrounding a common soft-iron core
• Turn ratio: Secondary has significantly more turns than primary (often thousands more)
• Voltage relationship: Voltage ratio follows the turns ratio (Vs/Vp ≈ Ns/Np)

The configuration amplifies voltage because:

• The shared soft-iron core maximizes magnetic coupling between coils
• When primary current rapidly changes, all secondary turns experience the changing flux
• The laminated core construction minimizes energy loss through eddy currents
• The large number of secondary turns means each small EMF per turn accumulates into a very high total voltage

This allows a low-voltage battery in the primary circuit to produce sparks between terminals G₁ and G₂ with potential differences of thousands of volts.

An induction coil transforms low-voltage DC to high-voltage pulses through:

1. Electromagnetic induction: Changing current in primary creates changing magnetic flux

2. Rapid current interruption: Mechanical "make and break" system creates asymmetric current changes (slow rise, rapid fall)

3. Rate enhancement: Capacitor accelerates current decay rate and creates current reversal, maximizing di/dt

4. Turn ratio advantage: Secondary coil typically has many more turns than primary (NS >> NP)

5. Flux concentration: Iron core maximizes magnetic flux linkage between coils

The combination of these factors allows a 12V DC source to produce pulses exceeding 50,000V, with voltage gain primarily determined by the turn ratio and the enhanced rate of change in the primary current.