Electromagnetic Induction Study Notes

Definition: Electromagnetic induction is the phenomenon in which an electric current or electromotive force (emf) is produced in a circuit by varying magnetic fields.
Experimental Foundations:
- Faraday and Henry's Experiments: Relative motion between a magnet and a conducting coil, or between two coils, induces an electric current in the coil.
- Transient Induction: A time-varying current in a stationary primary coil induces a momentary current in an adjacent secondary coil during circuit make or break.

When a bar magnet is pushed towards a coil, current is induced

Definition: For a planar surface of area $A$ placed in a uniform magnetic field &mathbf{B}, magnetic flux Φ_B is: Φ_B = &mathbf{B} ċ &mathbf{A} = B A &cos;(θ) where θ is the angle between the magnetic field &mathbf{B} and the area vector &mathbf{A}.
General Formulation: For curved surfaces or non-uniform fields: Φ_B = ∑ &mathbf{B}_i ċ d&mathbf{A}_i
Key Properties:
- Magnetic flux is a scalar quantity.
- SI Unit: Weber ( $Wb$ ) or tesla meter squared ( $T\,m^2$ ).

Law Statement: The magnitude of the induced emf in a circuit equals the time rate of change of magnetic flux through the circuit.
Mathematical Expressions:
- Single loop: ϵ = -&frac;{dΦ_B}{dt}
- Closely wound coil of $N$ turns: ϵ = -N &frac;{dΦ_B}{dt}
Flux Variation Methods: Flux can be changed by varying the magnetic field magnitude $B$ , modifying the coil area $A$ , or changing the relative orientation angle θ.

Lenz's Law Statement: The polarity of the induced emf is such that it produces a current that opposes the change in magnetic flux that caused it.
Energy Conservation: Mechanical work performed against the opposing magnetic force during motion is transformed into electrical energy, which dissipates via Joule heating.

Illustration of Lenz's law showing direction of induced current

Translational Motion: For a straight conductor of length $l$ moving at speed $v$ perpendicular to a uniform magnetic field $B$ : ϵ = B l v
Physical Derivation: Arises directly from the magnetic Lorentz force (&mathbf{F} = q &mathbf{v} × &mathbf{B}) acting on free charges within the moving conductor.
Rotational Motion: For a metallic rod of radius $R$ rotating at constant angular speed ω in a perpendicular uniform field $B$ : ϵ = &frac;{1}{2} B ω R^2

Setup showing motional emf with arm PQ sliding to the left

General Principle: Flux linkage N Φ_B is proportional to current $I$ (N Φ_B ∝ I). Inductance is the constant of proportionality depending strictly on geometry and medium permeability.
Properties: Scalar quantity with SI unit henry ( $H$ ) and dimensions $[M L^2 T^{-2} A^{-2}]$ .

Mutual Inductance ( $M$ ):
- Flux linkage in coil 1 due to current in coil 2: N_1 Φ_1 = M_{12} I_2
- Reciprocity relation: $M_{12} = M_{21} = M$
- For two long coaxial solenoids of length $l$ and inner radius $r_1$ : M = μ_0 n_1 n_2 π r_1^2 l
- Induced emf in secondary coil: ϵ_1 = -M &frac;{dI_2}{dt}
Self-Inductance ( $L$ ):
- Flux linkage in a single isolated circuit: N Φ_B = L I
- Self-induced back emf: ϵ = -L &frac;{dI}{dt}
- Long solenoid self-inductance: L = μ_0 n^2 A l
- Stored magnetic energy: W = &frac;{1}{2} L I^2
- Magnetic energy density in space: u_B = &frac;{B^2}{2 μ_0}

Operating Principle: Converts mechanical energy into alternating electrical energy through coil rotation inside a uniform magnetic field.
Induced EMF Formulation: For an armature coil of $N$ turns and area $A$ rotated at constant angular speed ω: Φ_B = B A &cos;(ω t) ϵ = N B A ω &sin;(ω t) = ϵ_0 &sin;(ω t) where ϵ_0 = N B A ω is the peak voltage.
Commercial Frequencies: Commercial generator rotation frequency is $50\,Hz$ in India and $60\,Hz$ in USA.