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The capacitor (2013)

Page history last edited by Joe Redish 6 years, 1 month ago

Working Content

 

Prerequisites:

 

In our analysis of two parallel sheets of equal amounts of opposite charge (Two parallel sheets of charge ) we learned that the field was effectively constant and restricted to the region between the two sheets.

 

Indeed, thinking more broadly about this result, it is not surprising.  Matter is made up of positive and negative charges and since they attract each other strongly, mostly they are near each other and cancel each other out. As a result, we typically don't see anything but neutral matter. If we imagine starting from a single sheet with positive and negative charges, we need to do work to pull the positive and negative charge sheets apart. In the pulling apart process, we have to do work on the charges and therefore build up electrical potential energy. We can therefore think of our two-sheet configuration as a way to store electrical energy as a separation of charge. Let's see how this works.

 

The capacitance

Let's suppose we have two sheets of equal and opposite charge that are quite close together. Suppose each sheet has an area A, they have opposite charges, +Q and -Q, that are spread uniformly, and they are a (small) distance apart, d. With this system, we can model the sheets as infinite and conclude that, except near the edges, the field in between the plates will be 

 

E = 4πkcσ = 4πkcQ/A.

 

Since we are interested in the energy, we want to re-express this relationship in terms of the potential difference between the two plates.

 

If we draw an x-axis perpendicular to the plates with its origin at the blue (positive) plate and the positive direction towards the red one, then the E field (x component) will be a positive constant between the plates and 0 outside. The graph will look something like the graph shown at the right. (We are ignoring the discrete nature of the charges. This will only show up if we get really close to a single charge.)

From this, we can infer the potential difference between the sheets in one of two ways.

 

First, we know that the potential energy difference can be calculated from the work needed to move the sheets apart, which is simply force times distance moved.  This is the application of the work-energy theorem for the electric force and PE. Now all we have to do is divide out the test charge on both sides of the equation.  Dividing out the test charge from the electric potential energy gives us the potential V.  To divide out the test charge from the work we simply have to divide the force by the test charge and end up with the electric field.

 

So, with the charge of the test charge divided out the work-energy theorem becomes a relation between a potential and a field:

where i refers to the starting (initial) point and f to the ending (final) point. Since is a constant between the plates, if we start at 0 and integrate along the x direction the integral becomes simple:

Alternatively, we can use that the E field is the gradient of the potential. Since we only have an x component, we have

This gives the same result -- not a surprise since the derivative is the opposite of the integral and they are representing two aspects of the same relationship between the field and the potential.

 

As a result, we can replace the E field, E, by ΔV/d to get:

 

We've ignored the sign here and are just representing a relation between magnitudes. That's because there is both a +Q and -Q (for a total charge equal to 0) and the sign of ΔV depends on which side we take as initial and which as final. So the sign really depends on how we choose to describe it. But the magnitude relationship holds no matter what choices we make.

 

This relation tells us that the charge separation on the sheets is associated with a potential difference. Since in many situations we will be controlling the potential difference (say with a battery or a chemical gradient) and we want to see how much charge separation we get, we'll express Q as something times the potential difference:

Our capacitor equation, Q = CΔV, tells us how much potential difference we need in order to get a charge separation of Q and -Q on the two sheets. The constant C is a property of the geometry of the sheets -- their area and separation -- and is called the capacitance.

 

The energy stored in a capacitor

The work you have to do to charge a capacitor by moving charges from one side of the plate to the other is equal to the amount of energy stored in it. This energy can be extracted to move charges around, light bulbs, or send signals down an axon. You are guided to work out the result in the homework problem, The Energy Stored in a Capacitor. The result is:

This makes good sense unit-wise, since we know voltage is energy per unit charge, so to get an energy we have to multiply by a charge -- and these are the charge and voltage difference we have. All you really have to do is figure out why there's a factor of 2.

 

This can also be expressed in other ways by using our capacitor equation:

 

We can see this energy in two ways; either as the amount of energy that is stored in the capacitor (useful in thinking about electrical devices) or as the amount of energy we need to put in in order to maintain a particular charge separation (useful in thinking about the charge separation across a membrane). We'll consider both examples in later readings and problems.

 

 Workout

 

Joe Redish 2/20/12

Wolfgang Losert 2/22/13

 

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