26.4-26.6
Ok we are now ready to put all the things together and apply them to circuits.
The book starts nicely by first introducing the battery and the the capacitor.
Not many examples or problems here in this section.
EMF
This section introduces the first two circuits concepts, the emf (pronounce
e-m-f) of a battery and the capacitance of capacitor.
EMF: This stands for electromotive force. Essentially it is a measure of the
work (energy) per unit of charge that a particular source can do. The precise nature
of the process depends on the source and typical battery use chemical
reactions to move positive ions. But later in this course, we will create emf
using magnetic fields. For a typical battery, the emf of the battery is nearly
the same as the voltage of the battery. For an ideal battery, it is the same.
You should assume ideal batteries in all physics problems unless otherwise
specified. In the video below I create an extremely non-ideal battery.
Capacitance.
Capacitance measures how hard it is to charge a particular object. It measures
the amount of charge you get per amount of potential it cost you.
Interestingly capacitance only depends on the shape of the object. This is
because different object shape will have different E field configuration (e.g.
line of charge versus plane of charge). The E field configuration determines
the potential (via the integral relationship).
The key formula is
C=\frac{Q}{\Delta V_C}
The book then discusses the equivalent capacitance of capacitor in series and
in parallel. See my video next page for more on this.
Energy in capacitor and E field
Formula 26.25 for the energy inside of a capacitor is super important. It is
obtained by calculating the total work of charging a capacitor and this is an
integral with the answer quadratic in the charge.
Conceptually, this is because it takes more energy to put an extra charge on a
charged capacitors than the energy it took to put the first charge at the
beginning.
Note that there is two ways to write this energy. Either as
U_C = \frac{Q^2}{2C}
or
U_C = \frac12 C(\Delta V_C)^2
This is potential energy, we can get back this energy by discharging the
capacitor. Where is the energy?? It turns out that the energy is in the
electric field. The electric field, again, is very real and contain the
energy. Eq 26.27 about the energy density of an electric field is always true.
Whenever there is an electric field in space, the energy density is given by
\frac{\epsilon_0}{2} E^2
. The total energy is this
density times the volume of space over which the E field constant. For a
non-constant E field, we need to do a volume integral.