Electromagentic Induction

Faraday’s Law

• Faraday’s law of electromagnetic induction states that a changing magnetic field induces an electromotive force (EMF) in a conductor. (NCERT Class 12, Chapter 6, Electromagentic Induction)

$$\text{emf}=-\frac{d\phi_B}{dt}$$ Where,

• (\text{emf}) is electromotive force

• (\phi_B) is magnetic flux

• The magnitude of the EMF is equal to the rate of change of the magnetic flux through the conductor.

• Lenz’s law determines the direction of the EMF and current produced by the induced magnetic field. (NCERT Class 12, Chapter 6, Electromagentic Induction)

• The direction of the induced current is such that the magnetic field created by the current opposes the change in the original magnetic field.

Self-Inductance

• Self-inductance of a coil is the property by virtue of which it opposes the change in current flowing through it by inducing an emf in itself. (NCERT Class 12, Chapter 7, Alternating Current)
• The self-inductance of a coil ((L)) is defined as the ratio of the magnitude of the magnetic flux ((\phi)) linked with the coil to the current (I) flowing through it. $$L=\frac{\phi}{I}$$
• SI unit of self inductance is Henry (H).
• The self-inductance of a solenoid of length ((l)), area of cross-section ((A)) containing (N) closely packed turns is given by: $$L=\frac{\mu_0N^2A}{l}$$ -Where (\mu_0) is the permeability of free space, having the value (4\pi\times10^{-7}\text{ T m/A or H/m}).
• The self-inductance of a toroid is given by $$L=\frac{\mu_0N^2}{2\pi R}$$ Where (R) is the mean radius of the toroid.

Energy Stored in a Magnetic Field

• When an inductor opposes the change in current flowing through it, some of the electrical energy is stored in the magnetic field associated with the inductor.
• The energy stored in a magnetic field ((U_B)) in a coil is given by $$U_B =\frac{1}{2}LI^2$$
• Where (I) is the current flowing through the inductor.

Inductors

• Inductors are electrical components designed to have high inductance.
• Inductors can be classified into two types based on the material used for the core:
• Air-core inductors: They use air as the core material.
• Iron-core inductors: They use iron as the core material. Iron-core inductors have a higher inductance compared to air-core inductors.
• Inductors in series: When inductors are connected in series, the total inductance of the circuit is the sum of the individual inductances. $$L_{eq}=L_1+L_2+…..+L_n$$
• Inductors in parallel: When inductors are connected in parallel, the total inductance of the circuit is given by $$ \frac{1}{L_{eq}}=\frac{1}{L_1}+\frac{1}{L_2}+\frac{1}{L_n}$$
• Energy stored in an inductor: The energy stored in an inductor is given by $$ U_L=\frac{1}{2}LI^2$$ where( I ) is the current flowing through the inductor.

RL Circuits

• When an inductor and a resistor are connected in series, it is called an RL circuit.
• Time constant of an RL circuit (({\tau})): The time constant of an RL circuit is the time taken by the current to reach (1-1/e\times100%\approx63.2%) of its steady-state value when the circuit is switched ON. $$$$\tau=\frac{L}{R}$$
• Transient analysis of RL circuits includes studying the behavior of the circuit when it is switched ON or OFF. - During charging , i.e., when the circuit is switched ON, the current rises exponentially with time according to the relation $$i(t)=I_0(1-e^{-t/\tau})}$$ $$- At t=\tau, i=\frac{I_0}{e}[[almost 63% of its final value]$$
• During discharging , i.e., when the circuit is switched OFF, the current falls exponentially with time as$$ i(t)=I_0\ e^{-t/\tau}$$ $$-At ( t=\tau, i=\frac{I_0}{e}[[almost 37% of its initial value]$$

Applications of Self-Inductance and Electromagnetic Induction

• Transformers: Transformers are devices that use the principles of self-induction and electromagnetic induction to transform alternating current (AC) voltage from one level to another. They consist of two coils, a primary coil, and a secondary coil, wound around a common iron core. The AC voltage applied to the primary coil creates a changing magnetic field in the core, which in turn induces an alternating EMF in the secondary coil. The ratio of the number of turns in the primary coil to the number of turns in the secondary coil determines the voltage transformation ratio of the transformer.

• Motors and generators: Motors and generators are electromechanical devices that convert electrical energy into mechanical energy and vice versa, respectively. Both motors and generators rely on the principles of electromagnetic induction. In a motor, an electric current flowing through a coil of wire creates a magnetic field. This magnetic field interacts with the magnetic field of a permanent magnet, causing the coil to rotate. In a generator, the reverse process occurs: the mechanical rotation of a coil of wire in a magnetic field induces an electric current in the coil.

• Magnetic resonance imaging (MRI): MRI is a medical imaging technique that uses the principles of electromagnetic induction to create detailed images of the inside of the body. MRI machines use powerful magnets to create a strong magnetic field, which aligns the spins of protons in the body’s water molecules. Radio waves are then pulsed into the body, causing the protons to flip their spins. When the radio waves are turned off, the protons return to their original spins, emitting electromagnetic waves. These waves are detected by the MRI machine and used to create images of the body’s tissues.