Electromagnetism & Modern Physics – Unit II (Magnetism) Study Notes

Electromagnetic Induction: General Ideas

1. Electromagnetic induction: generation of an emf whenever the magnetic flux linked with a closed conducting loop changes.

2. Induced emf / induced current: the emf and current that arise solely because of the flux change; disappear as soon as flux becomes steady.

3. Mathematically (qualitative statement of Faraday’s law)
   $\varepsilon = -\dfrac{d\Phi<em>B}{dt}$ where $\Phi</em>B$ is the magnetic flux linked with the circuit.

Faraday’s First Experiment (Magnet–Coil System)

• Apparatus: single coil connected to a sensitive galvanometer; bar‐magnet used as the external source of magnetic field.

• Observations

  • No deflection when magnet & coil are stationary → no flux change → no induced current.

  • Deflection occurs only while the magnet is in motion relative to the coil.
    • Magnet pushed towards coil (north pole leading) → pointer deflects in one direction.
    • Magnet pulled away → pointer deflects in opposite direction.
    • Replacing the north pole with the south pole reverses the sense of deflection for identical motions.

  • Deflection magnitude ∝ speed of relative motion (faster motion ⇒ larger magnitude of induced current).

• Key conclusion: relative motion between magnet & coil, not their absolute motion, is the cause of induction.

Faraday’s Second Experiment (Two‐Coil System)

• Replacement: bar magnet → second current-carrying coil (primary) connected to a battery → produces a steady magnetic field.

• The original galvanometer-coil behaves as the secondary coil.

• Observations analogous to the first experiment:

  • Moving primary coil towards secondary → galvanometer deflection; moving away → opposite deflection.

  • Greater speed of approach/withdrawal ⇒ larger deflection.

• Physical analogy: changing magnetic flux through the secondary due to motion of the primary is equivalent to flux change produced by a moving permanent magnet.

Lenz’s Law

• Statement: The induced current flows in such a direction that the magnetic field it produces opposes the change in magnetic flux that produced it.

• Qualitative form of energy conservation:

  • Opposition → a resistive force on the moving magnet/coil.

  • External mechanical work is converted → electrical energy → thermal (Joule) losses in the circuit.

• Applications/Devices exploiting Lenz’s law:

  • Eddy‐current balances, eddy‐current dynamometer.

  • Braking systems on trains, induction‐type AC generators.

  • Metal detectors, card readers, microphones, etc.

Fleming’s Hand Rules

Fleming’s Right-Hand Rule (Generators / Induced Current)

• Thumb (motion), forefinger (magnetic field $\vec B$), middle finger (induced current $I$) are mutually orthogonal.

• Useful for predicting direction of induced current in a moving conductor within a magnetic field (e.g., generator action).

Fleming’s Left-Hand Rule (Motors / Magnetic Force)

• Thumb (force $\vec F$), forefinger (field $\vec B$), middle finger (conventional current $I$) are mutually orthogonal.

• Determines direction of mechanical force on a current-carrying conductor placed in a magnetic field (motor action).

Magnetic Field Produced by Currents

Oersted’s Discovery (1820)

• Current in a straight wire deflects a nearby magnetic compass needle → proof that electric current produces a magnetic field.

• Reversing current reverses needle deflection → field direction depends on current direction.

• Unified electricity & magnetism → birth of electromagnetism.

Straight Current-Carrying Conductor

• Field lines: infinite series of concentric circles centered on the wire’s axis.

• Direction given by Right-Hand Thumb Rule: thumb = current, curled fingers = $\vec B$.

Circular Coil Carrying Current

• Near points on the loop: field lines still circular.

• Near the coil’s centre: field lines nearly parallel → almost uniform field inside a small region about the centre.

Maxwell’s Right-Hand Cork-Screw Rule

• Turn a right-handed screw so that it advances in the direction of current → rotation of the screw head gives direction of magnetic field lines.

Biot–Savart Law

• Differential contribution to magnetic field from a current element:
  $d\vec B = \frac{\mu_0}{4\pi}\;\frac{I\,d\vec s \times \hat r}{r^2}$

• Valid for steady currents in conductors as well as moving charge distributions (e.g., electron beam in TV CRT).

• Integrate over entire conductor to obtain net $\vec B$.

Magnetic Force Between Two Parallel Conductors

• Force per unit length:
  $\frac{F}{L} = \frac{\mu<em>0}{2\pi}\;\frac{I</em>1 I_2}{r}$
  where $r$ = separation of wires.

• Same current direction ⇒ attractive; opposite directions ⇒ repulsive.

Ampère’s Circuital Law

• Integral form:
  $\oint<em>{C} \vec B \cdot d\vec l = \mu</em>0 I_{\text{enclosed}}$

• Integral is path-independent provided path encloses the same net current.

• Useful for highly symmetric current distributions (solenoid, toroid, infinite wire, etc.).

Inductors and Inductance

Concept & Device

• Inductor = passive component (usually a coil) that stores energy in its magnetic field.

• Energy stored:
  $U = \tfrac12 L I^2$

• Inductance $L$ describes opposition to change of current; units: henry (H).

Classification by Core Material

• Iron-core, air-core, iron-powder, ferrite-core (soft vs hard ferrite) inductors.

• Symbol: $\begin{array}{c}\text{coil–like schematic symbol}\end{array}$.

Self-Inductance

• Changing current in a coil changes its own linked flux → induces emf in same coil.

• Magnitude:
  $\varepsilon<em>{\text{self}} = -L\,\frac{dI}{dt}, \qquad L = N\,\dfrac{\Phi</em>B}{I}$

Mutual Inductance

• Time-varying current in one coil (primary) induces emf in neighbouring coil (secondary).
  $\varepsilon<em>{\text{mutual}} = -M\,\frac{dI</em>1}{dt}$ where $M$ = mutual inductance.

Series & Parallel Inductor Combinations

• Series: equivalent inductance adds directly
  $L<em>{\text{eq}} = \sum</em>i L_i$

• Parallel: reciprocal addition
  $\frac{1}{L<em>{\text{eq}}} = \sum</em>i \frac{1}{L_i}$

• Current division: branch with smaller inductance carries larger dynamic current changes.

• Applications: power filters, tuned circuits, transformers, radio RADAR front-ends.

Eddy Currents

• When bulk conductor (plate/sheet) experiences changing magnetic flux, induced currents circulate in closed loops inside the material (no separate wire path).

• Called eddy (or Foucault) currents → resemble water eddies.

• Consequence: significant I²R heating; undesirable in transformer cores, but exploited in technology.

• Mitigation: laminate magnetic cores, use high-resistivity ferrites, slot the conductor, etc.

• Applications: induction cooktops, eddy-current brakes (trains, roller coasters), non-destructive testing, electromagnetic damping (galvanometers).

Transformer

• Static AC machine: transfers electrical power between two circuits by electromagnetic induction; frequency unchanged.

• Types by voltage conversion
  • Step-up: Vs > Vp (secondary turns > primary turns).
  • Step-down: Vs < Vp.

• Working principle: mutual induction between primary & secondary coils wound on common magnetic core.

• Fundamental equation for ideal transformer (neglecting losses):
  $\frac{V<em>s}{V</em>p} = \frac{N<em>s}{N</em>p} = \frac{I<em>p}{I</em>s}$

• Energy conversion chain: AC source → changing primary current $I<em>p$ → alternating magnetic flux in core → induced emf $V</em>s$ in secondary → load current.

Core & Copper Losses

• Core (iron) loss = hysteresis + eddy current losses.

• Copper loss = $I^2 R$ heating in windings.

• Leakage flux = flux not linking both coils; reduces coupling coefficient.

• Transformers have no moving parts ⇒ high efficiency (95–99%) but non-zero losses.

Limitation

• Cannot operate with DC: constant current produces steady flux → no induced emf in secondary; quickly saturates core & overheats.

Eddy-Current & Lenz-Law-Based Devices (Quick List)

• Eddy current balances/dynamometers.

• Induction stove, magnetic braking, metal detectors, card readers, microphones, AC generators.

Summary Connections & Energy Perspective

• Faraday’s discovery → link between electric circuits & moving magnets; quantified by laws of induction.

• Lenz’s law ensures energy conservation by making induced currents oppose the causative flux change.

• Fleming’s hand rules provide the right-angle triads connecting field, current, and motion/force.

• Biot–Savart & Ampère’s laws together play the same role for magnetostatics as Coulomb & Gauss laws do for electrostatics.

• Inductance & transformers exploit the storage & transfer aspects of magnetic energy in coiled conductors.

• Eddy currents illustrate both useful (braking, heating) and undesirable (core losses) facets of induction.