Comprehensive Study Notes on Inductors, Faraday's Laws, and Electromagnetic Principles

Energy Storage Comparison:
- Capacitors store energy in an electric field.
- Inductors store energy in a magnetic field.
Faraday's First Law of Electromagnetic Induction:
- When a coil is placed in a time-varying magnetic field, an electromotive force (voltage) is induced across the coil.
- Relative motion between a magnetic field and a conductor produces a dynamic induced potential difference across the terminals of the coil.
Faraday's Second Law of Electromagnetic Induction:
- The magnitude of the induced voltage is directly proportional to the time rate of change of magnetic flux linking the coil.
- Mathematical proportionality: $\text{Induced Voltage} \propto \frac{d\phi}{dt}$
- Formulation for a coil having $N$ turns: $v = N \frac{d\phi}{dt}$
- Here, $N$ represents the number of turns of the winding, and $\phi$ represents the magnetic flux measured in Webers ( $\text{Wb}$ ).

Lenz's Law:
- The direction of the induced current (or induced voltage) caused by a changing magnetic field in a conductor always creates its own magnetic field that opposes the original change in magnetic field that produced it.
- Incorporating the opposing polarity determined by Lenz's Law gives the voltage relation: $v = -N \frac{d\phi}{dt}$
Concept of Self-Inductance ( $L$ ):
- Self-inductance relates the voltage induced in an electrical circuit to the time rate of change of electric current through that same circuit: $v = L \frac{di}{dt}$
- Equating the electromagnetic voltage expression with the circuit inductance expression: $L \frac{di}{dt} = N \frac{d\phi}{dt}$
- Expressing inductance in differential form: $L = N \frac{d\phi}{di}$
- For linear media where flux is proportional to current: $L = \frac{N \phi}{i}$
- Unit of inductance: Henry ( $\text{H}$ ).
Magnetic Flux Linkage ( $\psi$ ):
- Flux linkage represents the total magnetic flux passing through all turns of a coil: $\psi = N \phi$
- Relation to inductance and current: $L i = N \phi = \psi$
- Units:
- Magnetic flux ( $\phi$ ) is measured in Webers ( $\text{Wb}$ ).
- Flux linkage ( $\psi$ ) is measured in Weber-turns ( $\text{Wb-turns}$ ).

Derivation of Inductor Current:
- Starting with the fundamental terminal equation: $v = L \frac{di}{dt}$
- Rearranging terms to isolate the differential current: $\frac{di}{dt} = \frac{v}{L}$ $di = \frac{v}{L} \, dt$
- Integrating both sides over the time interval from $0$ to $t$ : $\int_{i(0)}^{i(t)} di = \frac{1}{L} \int_0^t v \, dt$ $i(t) - i(0) = \frac{1}{L} \int_0^t v \, dt$ $i(t) = \frac{1}{L} \int_0^t v \, dt + i(0^+)$
- Here, $i(0^+)$ denotes the initial current established in the inductor at $t = 0^+$ .
Initial Charging Condition:
- If the inductor is initially discharged (possesses zero pre-existing magnetic energy), then: $i(0^+) = 0$
- The complete current equation simplifies to: $i(t) = \frac{1}{L} \int_0^t v \, dt$
Instantaneous Power in an Inductor:
- Electrical power delivered to the inductor is defined by the product of terminal voltage and current: $p = v \times i$
- Substituting the terminal voltage relationship $v = L \frac{di}{dt}$ : $p = L i \frac{di}{dt}$
Stored Magnetic Energy ( $E$ ):
- The total energy stored in the inductor's magnetic field over time is the integral of instantaneous power: $E = \int p \, dt$ $E = \int \left(L i \frac{di}{dt}\right) dt$ $E = \int L i \, di$
- Carrying out the integration yields: $E = \frac{1}{2} L i^2 \, \text{Joules}$

Reluctance of a Magnetic Circuit ( $S$ ):
- Inductance can be defined through the magnetic reluctance of the core material: $L = \frac{N^2}{S}$
- Magnetic reluctance $S$ is defined by the physical dimensions and material property: $S = \frac{l}{\mu A}$
- $S$ : Reluctance of the magnetic core
- $l$ : Length of the coil or magnetic path
- $A$ : Cross-sectional area of the coil
- $N$ : Total number of turns of the winding
- $\mu$ : Absolute magnetic permeability of the core material
Formula for Geometric Inductance:
- Substituting the expression for reluctance $S$ into the inductance equation: $L = \frac{N^2}{\frac{l}{\mu A}}$ $L = \frac{N^2 \mu A}{l}$
- Measured in Henry ( $\text{H}$ ).
Parametric Proportionalities:
- Inductance is directly proportional to the square of the number of turns: $L \propto N^2$
- Inductance is directly proportional to the magnetic permeability: $L \propto \mu$
- Inductance is directly proportional to the cross-sectional area: $L \propto A$
- Inductance is inversely proportional to the coil length: $L \propto \frac{1}{l}$

Absolute Permeability ( $\mu$ ):
- Denoted in Hindi as पारगम्यता (Paargamyata).
- Defined as the inherent property of a material indicating its capability to store energy in the form of a magnetic field and permit magnetic flux lines to establish through it.
- Measured in units of Henry per meter ( $\text{H/m}$ ).
Mathematical Breakdown:
- Absolute permeability is the product of the permeability of free space and the relative permeability of the specific medium: $\mu = \mu_0 \mu_r$
Permeability of Free Space ( $\mu_0$ ):
- Constant value for vacuum or free space: $\mu_0 = 4\pi \times 10^{-7} \, \text{H/m}$
Relative Permeability ( $\mu_r$ ):
- A dimensionless, unitless ratio comparing material permeability to that of free space: $\mu_r = \frac{\mu}{\mu_0}$

Problem Statement:
- A constant current of $5 \, \text{A}$ is flowing through a $10 \, \text{H}$ inductor. The voltage induced in the inductor will be:
- a) $50 \, \text{V}$
- b) $500 \, \text{V}$
- c) $2 \, \text{V}$
- d) $0 \, \text{V}$
Given Parameters:
- Current: $i = 5 \, \text{A} = \text{constant}$
- Inductance: $L = 10 \, \text{H}$
Analytical Calculation:
- Applying the voltage-current relationship: $v = L \frac{di}{dt}$
- Because the current is constant over time, its first derivative is zero: $\frac{d(5)}{dt} = 0$
- Computing terminal voltage: $v = 10 \times \frac{d(5)}{dt}$ $v = 10 \times 0 = 0 \, \text{V}$
Conclusion:
- The induced voltage across the inductor is $0 \, \text{V}$ . The correct option is d.