Rotating Coils: Sinusoidal EMF Generation in Electric Generators
How Does a Rotating Coil Generate EMF?
- Consider a coil rotating in a magnetic field.
- As it rotates, the magnetic flux through the coil changes, inducing an emf according to Faraday’s law.
The induced emf is proportional to the rate of change of magnetic flux.
Mathematical Description of Sinusoidal EMF
- If a coil with $N$ turns and area $A$ rotates in a uniform magnetic field $B$ with angular velocity $v$, the magnetic flux $\Phi$ through the coil is given by: $$
\Phi = NBA\cos(\omega t)
$$ - The induced emf is the negative rate of change of this flux: $$
\varepsilon = -\frac{d\Phi}{dt} = NBA\omega\sin(\omega t)
$$ - This equation describes a sinusoidal emf, which oscillates between positive and negative values.
The peak emf is given by $\varepsilon_0=NBA\omega$, and the emf varies as $\varepsilon =\varepsilon_0 \sin{(wt)}$.
Why Sinusoidal?
- The sinusoidal nature arises because the flux changes cyclically as the coil rotates.
- When the coil is perpendicular to the field, the flux is maximum (but the rate of change is zero), resulting in zero emf.
- When the coil is parallel to the field, the flux is zero, but the rate of change is maximum, producing maximum emf.
- Think of the coil’s rotation like a pendulum swinging back and forth.
- At the extremes, the pendulum is momentarily still (zero emf), but at the midpoint, it moves fastest (maximum emf).
Time-Varying Magnetic Fields: Induction by Moving Magnets or Coils
How Does Motion Induce EMF?
EMF can be induced by:
- Moving a magnet near a coil.
- Moving a coil in a magnetic field.
- Changing the strength of the magnetic field.
- When a magnet is pushed into a coil, the galvanometer needle deflects, indicating a current.
- Pulling the magnet out causes the needle to deflect in the opposite direction.
Factors Affecting Induced EMF
- Speed of Motion: Faster movement increases the rate of change of flux, inducing a larger emf.
