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\PhiB}{dt} where \PhiB 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{\mu0}{2\pi}\;\frac{I1 I_2}{r}
    where r = separation of wires.

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

Ampère’s Circuital Law

  • Integral form:
    \oint{C} \vec B \cdot d\vec l = \mu0 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{\text{self}} = -L\,\frac{dI}{dt}, \qquad L = N\,\dfrac{\PhiB}{I}

Mutual Inductance

  • Time-varying current in one coil (primary) induces emf in neighbouring coil (secondary).
    \varepsilon{\text{mutual}} = -M\,\frac{dI1}{dt} where M = mutual inductance.

Series & Parallel Inductor Combinations

  • Series: equivalent inductance adds directly
    L{\text{eq}} = \sumi L_i

  • Parallel: reciprocal addition
    \frac{1}{L{\text{eq}}} = \sumi \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{Vs}{Vp} = \frac{Ns}{Np} = \frac{Ip}{Is}

  • Energy conversion chain: AC source → changing primary current Ip → alternating magnetic flux in core → induced emf Vs 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.