Electromagnetic Induction and Alternating Currents – JEE Mains Physics

1. Electromagnetic Induction: Faraday's Law

Faraday's law states that a change in magnetic flux through a coil induces an electromotive force (emf) in the coil. The induced emf is given by:
```
ε = -dΦ/dt
```
where ε is the induced emf and Φ is the magnetic flux.

The induced emf generates a current if the circuit is closed. According to Lenz’s law, the direction of the induced current opposes the change in flux that produced it, which ensures the conservation of energy.

Lenz’s law states that the direction of the induced current in a closed loop will be such that it opposes the change in the magnetic flux that induced it. This is a manifestation of the law of conservation of energy.

Eddy currents are circulating currents induced in conductors when they are exposed to a changing magnetic field. These currents can lead to energy loss in the form of heat and can be minimized by using laminated cores in electrical machines.

Self-inductance (L) is the property of a coil that opposes the change in current through it. The induced emf is related to the rate of change of current as:
```
ε = -L (di/dt)
```
Mutual inductance (M) is the property of two coils where a change in current in one coil induces emf in the second coil. The induced emf is related to the rate of change of current in the first coil as:
```
ε₂ = -M (di₁/dt)
```

Alternating current (AC) is the current that changes direction periodically. The standard equation for AC is:
```
I(t) = I₀ sin(ωt)
```
where I₀ is the peak current, ω is the angular frequency, and t is time.

The peak value (I₀) is the maximum value of the current, whereas the root mean square (RMS) value (Iₓ) is the effective value of the current, given by:
```
Iₓ = I₀ / √2
```
The RMS value is used to calculate power in AC circuits.

Reactance is the opposition to the flow of AC due to inductance (XL) and capacitance (XC). It is given by:
```
XL = ωL, XC = 1/(ωC)
```
Impedance (Z) is the total opposition to the flow of AC in a circuit and is given by:
```
Z = √(R² + (XL - XC)²)
```
where R is the resistance.

In an LCR series circuit, the total impedance is the sum of the resistive, inductive, and capacitive reactances. The resonance occurs when the inductive reactance equals the capacitive reactance, resulting in the lowest impedance.

Resonance occurs in an LCR circuit when the inductive reactance (XL) equals the capacitive reactance (XC). At resonance, the impedance is minimum, and the current is maximum. The resonance frequency (f₀) is given by:
```
f₀ = 1 / (2π√(LC))
```

The power in an AC circuit is given by:
```
P = Iₓ² R
```
where Iₓ is the RMS value of current and R is the resistance. The power factor (cos(φ)) determines how much of the total power is useful.

Wattless current is the current in a purely inductive or purely capacitive circuit, where no power is consumed. The current leads or lags the voltage by 90 degrees in these circuits.

An AC generator converts mechanical energy into electrical energy using electromagnetic induction. The output voltage varies sinusoidally with time.
A transformer works on the principle of mutual inductance and is used to step up or step down the AC voltage. The relationship between the primary and secondary voltages and currents is given by:
```
V₁ / V₂ = N₁ / N₂
```
where N₁ and N₂ are the number of turns in the primary and secondary coils, respectively.