Peter's Physics Pages

Physics for Civil Engineering

This is an introduction to Electricity, Strength of Materials and Waves.

Lecture 4 (Capacitors)

In this lecture the following are introduced:
• the capacitance of two conducting surfaces
• the effect of dielectrics (insulators) between the capacitor plates
• the series and parallel combination of capacitors
• the energy stored in a capacitor

Introduction

The capacity of something is its "holding power" - how much it can absorb, hold, or store. In electricity a "capacitor" is a device which stores electric charge.

Any conductor has surfaces, and charge can accumulate on those surfaces. This surface charge influences any nearby conductors, thus forming a capacitor.

 Two flat metal plates can form a simple capacitor, as shown below. When one plate is connected to the positive terminal of a battery and the other plate to the negative terminal, positive charge flows away to spread out over the first plate. Since +Q leaves the positive side of the cell and +Q returns to the negative side, it appears as if a current flows around the circuit, but no charge actually crosses between the plates. Charge only crosses the plates when the material between the plates has broken down and become conductive.

Capacitance

A capacitor (device) has capacitance (property) The symbols and units are For a parallel plate capacitor, the larger the area of the plates, the bigger the capacitance. Also, the closer together the plates can be placed without breakdown, the bigger the capacitance. Air or vacuum between the plates has an insulating effect. The proportionality constant for air/vacuum is ε0=8.85 pF.m-1. This constant is called the Electric permittivity of free space.

To keep the plates apart, an insulating or dielectric material (paper, oil, or mica), is often placed between the plates. This also dilutes the effect of the charges by reducing the potential difference between the plates compared to the same charge without the material. The dilution factor multiplies the capacitance. The multiplication is accounted for by using a multiplier called the dielectric constant, .

The capacitance of a parallel plate capacitor with a dielectric between the plates is given by: Dielectric constants at 200C

 Dielectric material Dielectric constant κ Dielectric strength* vacuum 1 infinite air 1.0006 0.8 kV/mm Ethanol 24 glass 4 - 7 hard rubber 2.8 mica 7 160 kV/mm paper 3 - 7 14 kV/mm paraffin 2.2 plastic 2.8 - 4.5 porcelain 6 - 8 pure water 80 Quartz 4.3 8 kV/mm Teflon 2.1 60 kV/mm Titanium dioxide 100 6 kV/mm

*The dielectric strength is the resistance to breakdown, i.e. the maximum allowed potential difference per distance between the plates before the material breaks down and forms a conducting path.

Example C14

A parallel plate capacitor has metal plates of area 26.9 m2. The plates are separated by Teflon that has a uniform thickness of 0.1 mm. The dielectric constant of Teflon is 2.1, and the dielectric strength is 60 kV/mm. Find the capacitance, and the maximum potential difference allowed.  The maximum potential difference allowed between the plates is 6kV.

Uses for Capacitors

1. To shape electric fields. E.g. to deflect electron beams in Cathode Ray Tube TVs
2. To store electrical energy in small volumes. E.g. like rechargeable batteries
3. To reduce Electric Potential fluctuations, to transmit pulsed signals, to generate radio waves, etc.
4. Parallel sheets of charge are found in biological cells and form capacitors essential to cellular activity.

Capacitors connected in series

 Capacitors are in series when (i) the same charge is stored on each, and (ii) they are connected + to -, as shown in the diagram. As with resistors in lecture 2, the + and - show the relative potentials across each element and not absolute potentials. The flow of charge in a series circuit

When a switch connects the e.m.f. to the circuit, there is an initial flow of charge. The following illustrates what happens.

 Charge +Q leaves the positive side of the cell and moves to the first side of C1 making it positive. The presence of positive charge +Q on the first side of the plate causes an equal charge +Q to leave the second side of C1 leaving -Q behind, making the total charge stored on C1 to be Q. The +Q leaving C1 moves to the first side of C2 making it positive. Then +Q leaves the second side of C2 leaving -Q behind, so the charge stored on C2 is also Q. The +Q leaving C2 moves to the first side of C3 making it positive. Then +Q leaves the second side of C3 leaving -Q behind, so the charge stored on C3 is also Q. Each capacitor has the same charge Q stored on it. However the total charge stored is not 3Q but Q, since Q leaves the cell and Q returns to it.

We have the unusual situation that the total charge stored on all 3 capacitors is also the charge stored on each capacitor. This is because the original neutral charge between adjacent capacitors is separated into plus and minus on different plates.

Equivalent capacitance for a series combination

When capacitors are connected in series they have the same charge stored on each, and the series approach is to add the potential differences.   Example C15

A 20μ, a 30μ, and an 8μ capacitor are connected in series with a 240V battery. Find the charge stored and potential difference across each capacitor.  Notice that, like resistors in parallel, the total is less than the smallest. The total charge stored is then Since the total charge stored is also the charge stored on each, each capacitor has 1152μF stored on it. Notice that the smallest capacitor has the largest potential difference across it.

Capacitors connected in parallel.

 Capacitors are in parallel when they (i) have the same potential difference across them, and (ii) are connected + to + and - to -, as shown in the diagram. Equivalent capacitance for capacitors connected in parallel

 When capacitors are connected in parallel they have the same potential difference across each, and the parallel approach is to add the charges stored.  Charge flows out from the e.m.f and divides proportionally with the capacitance.

Example C16
A 10μ, a 20μ, and a 30μ capacitor are connected in parallel with a 240V battery. Find the total charge stored in the circuit and the charge stored on each capacitor.   Adding these together gives the same total as before. The largest capacitor stores the largest charge.

Energy stored in a capacitor

From its definition To "charge" a capacitor, work has to be done. The first bit of charge ΔQ, has no potential difference to be worked against, but creates a small potential difference ΔV. The next bit of charge has to be worked against that small potential difference (ΔW = ΔV×ΔQ). As the plate becomes more positive and you have to work harder to get the next positive charge to flow onto it.

 Graphically, the work done against a potential difference is the area under the potential vs charge graph.  Using the definition of capacitance (C=Q/V) we can replace V or Q to get the energy stored, U Joule, in terms of Q and V, C and V, or Q and C. Example C17

 For each capacitor in the circuit shown, find (a) the potential difference across it, (b) the charge stored on it, and (c) the energy stored in it. First, reduce the parallel combination: C||=C2+C3=(5+25)μF=30μF The circuit becomes: Next, reduce the series combination: The circuit becomes: The total charge stored can now be calculated. Working back, the two capacitors in the series combination of 20μF and 30μF each have the total charge of 2.88μF stored on them. The potential differences are: Check: these should add to be 240V Working back, the circuit is now: The charges stored on the 5μF and 25μF in parallel are given by: Check: These should add to 2.88mC Now for the energy stored:  Check: The sum of the energies stored on the individual capacitors adds up to the correct total.

Summarising:

The dielectric strength is the resistance to breakdown, i.e. the maximum allowed potential difference per distance between the plates.

 Definition of capacitance: Parallel plate capacitance from construction Equivalent capacitance for capacitors in series Equivalent capacitance for capacitors in parallel Energy stored in a capacitor  email me a note if you found this useful
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