Faraday's law
Faraday's Law of Electromagnetic Induction
Faraday's law of electromagnetic induction is one of the fundamental principles of electromagnetism. It describes how a changing magnetic field can induce an electric current in a conductor. This principle is the working basis for many electrical devices, such as transformers, inductors, and generators.
Understanding Faraday's Law
Michael Faraday discovered in 1831 that a change in magnetic environment of a coil of wire will cause a voltage (emf - electromotive force) to be induced in the coil. The law has two aspects:
- Induction of EMF: An EMF is induced in a conductor when the magnetic flux linking with the conductor changes.
- Law of Lenz: The direction of the induced EMF is such that it tends to produce a current which opposes the change in magnetic flux that produced it.
Mathematical Formulation
Faraday's law can be mathematically expressed as:
$$ \mathcal{E} = -\frac{d\Phi_B}{dt} $$
where:
- $\mathcal{E}$ is the induced EMF (in volts)
- $\Phi_B$ is the magnetic flux through the circuit (in webers, Wb)
- $t$ is the time (in seconds)
- The negative sign is a representation of Lenz's law, indicating the direction of the induced EMF opposes the change in magnetic flux.
Magnetic Flux
Magnetic flux, $\Phi_B$, is a measure of the quantity of magnetism, taking into account the strength and the extent of a magnetic field. It is given by:
$$ \Phi_B = B \cdot A \cdot \cos(\theta) $$
where:
- $B$ is the magnetic field strength (in teslas, T)
- $A$ is the area of the loop (in square meters, m²)
- $\theta$ is the angle between the magnetic field lines and the normal (perpendicular) to the area $A$
Differences and Important Points
| Aspect | Description |
|---|---|
| Induced EMF | The voltage generated due to the changing magnetic flux. |
| Magnetic Flux ($\Phi_B$) | The product of the average magnetic field times the perpendicular area that it penetrates. |
| Lenz's Law | The direction of the induced EMF is such that it opposes the change in magnetic flux. |
| Magnetic Field (B) | The magnetic field in which the conductor is situated. |
| Area (A) | The area of the loop or coil through which the magnetic field passes. |
| Angle ($\theta$) | The angle between the magnetic field and the normal to the loop's area. |
Examples
Example 1: Single Loop
Consider a single loop of wire with an area of $0.02 \, m^2$ placed in a magnetic field that changes uniformly from $0.5 \, T$ to $0 \, T$ in $0.1 \, s$. The loop is perpendicular to the magnetic field ($\theta = 0^\circ$). Calculate the induced EMF.
Solution:
- Calculate the change in magnetic flux, $\Delta \Phi_B$:
$$ \Delta \Phi_B = B_f \cdot A - B_i \cdot A = (0 - 0.5 \, T) \cdot 0.02 \, m^2 = -0.01 \, Wb $$
- Calculate the induced EMF, $\mathcal{E}$:
$$ \mathcal{E} = -\frac{\Delta \Phi_B}{\Delta t} = -\frac{-0.01 \, Wb}{0.1 \, s} = 0.1 \, V $$
Example 2: Coil with Multiple Turns
A coil with 10 turns (N = 10) has an area of $0.1 \, m^2$ and is placed in a magnetic field that changes at a rate of $2 \, T/s$. The coil is aligned such that the magnetic field is parallel to the normal of the area ($\theta = 0^\circ$). Calculate the induced EMF.
Solution:
- Calculate the rate of change of magnetic flux, $\frac{d\Phi_B}{dt}$:
$$ \frac{d\Phi_B}{dt} = N \cdot A \cdot \frac{dB}{dt} = 10 \cdot 0.1 \, m^2 \cdot 2 \, T/s = 2 \, Wb/s $$
- Calculate the induced EMF, $\mathcal{E}$:
$$ \mathcal{E} = -N \cdot \frac{d\Phi_B}{dt} = -10 \cdot 2 \, Wb/s = -20 \, V $$
The negative sign indicates that the induced EMF will act to oppose the increase in magnetic flux.
Conclusion
Faraday's law of electromagnetic induction is a cornerstone of modern electromagnetism and is essential for understanding how electric generators, transformers, and other devices operate. It shows the relationship between a changing magnetic field and the electric current it can induce. By studying Faraday's law, we gain insight into the conversion of energy from magnetic to electrical forms and vice versa.