Charge Flown Due to Induced Current

Electromagnetic induction is a fundamental principle in physics that describes the process by which a changing magnetic field within a closed loop induces an electromotive force (EMF), leading to the flow of electric current if the loop is closed. This phenomenon was discovered by Michael Faraday and is described by Faraday's law of induction.

Faraday's Law of Induction

Faraday's law states that the induced electromotive force in any closed circuit is equal to the negative of the time rate of change of the magnetic flux through the circuit. Mathematically, it is expressed as:

$$ \mathcal{E} = -\frac{d\Phi_B}{dt} $$

where $\mathcal{E}$ is the induced EMF and $\Phi_B$ is the magnetic flux.

Lenz's Law

Lenz's law gives the direction of the induced current. It states that the induced current will flow in such a direction that it will oppose the change in magnetic flux that produced it. This is a manifestation of the conservation of energy.

Charge Flown Due to Induced Current

When an induced current flows through a conductor, charges are transported from one part of the conductor to another. The total charge $Q$ that flows due to the induced current $I$ over a time interval $t$ can be calculated using the formula:

$$ Q = I \cdot t $$

However, if the current is not constant over time, we must integrate the current over the time interval:

$$ Q = \int_{t_1}^{t_2} I(t) \, dt $$

where $I(t)$ is the instantaneous current at time $t$.

Example: Induced Current in a Solenoid

Consider a solenoid with $N$ turns, length $l$, and cross-sectional area $A$. If the magnetic field inside the solenoid changes at a rate of $\frac{dB}{dt}$, the induced EMF can be calculated using Faraday's law:

$$ \mathcal{E} = -N \frac{d\Phi_B}{dt} = -N A \frac{dB}{dt} $$

If the solenoid has a resistance $R$, then the induced current $I$ is given by Ohm's law:

$$ I = \frac{\mathcal{E}}{R} = -\frac{NA}{R} \frac{dB}{dt} $$

The charge flown over a time interval $t$ is:

$$ Q = -\frac{NA}{R} \int_{0}^{t} \frac{dB}{dt} \, dt = -\frac{NA}{R} (B(t) - B(0)) $$

Table of Differences and Important Points

Property	Induced EMF ($\mathcal{E}$)	Induced Current ($I$)	Charge Flown ($Q$)
Definition	EMF induced due to a changing magnetic field	Current resulting from the induced EMF	Total charge that has passed through the conductor due to the induced current
Formula	$\mathcal{E} = -\frac{d\Phi_B}{dt}$	$I = \frac{\mathcal{E}}{R}$ (if $R$ is constant)	$Q = \int I(t) \, dt$
Units	Volts (V)	Amperes (A)	Coulombs (C)
Dependence	Depends on the rate of change of magnetic flux	Depends on the induced EMF and resistance of the circuit	Depends on the induced current and the time interval
Direction	Given by Lenz's law	Given by Lenz's law and the polarity of the induced EMF	Direction is the same as the induced current

Conclusion

Understanding the charge flown due to induced current is crucial for analyzing and designing electrical systems and devices that operate based on electromagnetic induction. This includes transformers, electric generators, induction motors, and many other applications in modern technology. The principles of Faraday's law and Lenz's law are foundational to the study of electromagnetism and its practical applications.