DC Generators

Definition

An electrical component which provides either current or voltage. Either batteries (DC) or generators (AC/DC).

An ideal current generator provides constant current, while an ideal voltage generator provides constant potential difference between its terminals.

Real Generators are not 100% efficient. Below are equivalent circuits to a real voltage generator and a real current generator respectively:

• Real Voltage Generator: the lower the internal resistance, the better.
• Real Current Generator: the higher the internal resistance, the better.

EMF

Definition

Electromotive Force (EMF) is the work done per unit charge by an electric field generated by a time dependant magnetic flux.

Three options to generate it:

• Vary surface enclosed by the circuit (${\Phi }_{\vec{B}}\propto S$)
• Vary magnetic field over time
• Vary angle between field and surface over time (${\Phi }_{\vec{B}}\propto \cos \theta$)
  • $\text{emf}=BS\omega \sin \omega t$
  • $P=\tau \omega$

If $\vec{B}$ is uniform across a perpendicular surface $S$, we get:

$$\text{emf}=-\frac{d\vec{B}}{dt}S$$

If we increase the surface, by moving one side of length $L$ at velocity $v$, we get:

$$\text{emf}=BLv$$

And

$$F=ILB$$

Since we are moving charges through a magnetic field, they are affected by the Lorentz Force. HOWEVER, if we take the bar as a point of reference, the charges are stationary, so an electric field MUST be pushing them through the bar. Therefore,

$$\vec{F}=q(\vec{v}\times \vec{B})=q\vec{E}$$

So, we have:

$$\vec{E}=\vec{v}\times \vec{B}$$

And

$$\Delta V=EL=vBL=\text{emf}$$

Summing it all up

$$F=q(\vec{E}+\vec{v}\times \vec{B})$$