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# Capacitance

last edited by 4 years, 9 months ago

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Prerequisites:

An important reason for introducing the ideas of electric field and electric potential is that the electric field is so strong (compared to, say, gravity) and that everything is made of of electric charges. As a result, introducing an unbalanced charge into a system of objects can result in a lot of charge moving around. It would be impossible to keep track of all the charges and sum their forces.

But if we consider the electric fields and potentials produced by these charges, things become a lot simpler. Large scale constraints that follow from the properties of a system can control where all those charges go and produce a simple result. For example, if we put some charge on an object that can conduct electricity (charges can move freely on it), then for a static (charges are not moving) situation, we can easily show that the entire body of the conductor must all have the same electrostatic potential. This constrains what the electric field near the surface must be near the surface (it must be perpendicular to the surface) and relates the magnitude of that field to the magnitude of the charge density on this surface of the conductor.* So even though we can't typically count or even see individual charges on an object, if we can measure electric fields (or, equivalently, electrostatic potentials), we can infer a lot about how much charge we have and where it is.

The way that charges distribute themselves on a conductor has a lot to do with the shape of the conductor. When we put charge on a conductor, the charge will move around until it reaches the state with the lowest free energy. When that happens, there will be potential differences created. The way those potential differences are related to the charges that create them is called the capacitance.

For example, suppose we take a conducting sphere of radius R and put a charge, Q on it. Since the particles of which the charge is made up are of the same sign, they will repel and try to get as far apart as possible. As a result, the will all go to the surface of the sphere and spread out as uniformly as possible. We know from our discussion of spherical charges (A simple electric model: a sphere of charge) that the electric field outside the charge will be the same as for a point charge: E = kCQ/r2. This means that the electric potential at the surface of the conductor (taking the potential at infinity to be 0) will be V = kCQ/R. So the potential of the conductor is proportional to the charge on it. We write:

ΔV = CQ

where C is defined as the capacitance of the object. For the case of a single sphere compared to infinity, C = kC/R. We put in the delta to emphasize that we are describing a difference in potentials. For the conducting sphere, the difference is the potential at the surface of the sphere compared to the potential at infinity, but since we can take the potential at infinity to be 0, here it doesn't matter. In most cases, the "Δ" will be very important!

More typically, we will be interested in a potential difference created between two surfaces by equal and opposite charges placed on those two surfaces. (See Example: Two parallel plates.)

These distributions of charge store electric energy by keeping apart charges that would prefer to get together. In a typical two-sided capacitor that has a positive charge +Q on one side and a negative charge -Q on the other, the energy stored in the capacitor is

energy stored = ½QΔV

where ΔV  is the difference in the potential between the two sides. (It's not the "qV" we would expect, but only half. See: The energy stored in a capacitor.) If we know

Capacitance plays important roles in many systems in which charge moves and electrical energy needs to be stored for later use. The membranes of cells have opposite charges on either side and act as capacitors. Capacitors play a critical role in electrical oscillators, helping to tune systems to pick up radio and wifi signals.

* These sort of results follow from the mathematics of field theory -- equations that relate the fields and potentials to the charge distributions. Since these equations (Maxwell's equations of electromagnetism) belong to vector calculus, we won't be writing or solving them here.

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Joe Redish 2/18/16