THE PHYSICS BEHIND CHARGE AND FIELD: Electrical Potential, Capacito...

In using a flame probe for investigating the potential between two parallel plates, a voltmeter is used to measure the p.d. across the parallel plates. A flame probe is used to measure voltage at points in the space between the plates; the voltage increases as the probe is moved from the left plate, which is at 0 V, to the right plate. More precisely, the probe measures a voltage in the field between the plates. This voltage is called the electric potential. The potential can be defined as the p.d. between the plate at 0 V (or zero potential) and the probe.

^{Computed electrostatic equipotentials (black contours) between two electrically charged spheres. Zureks, CC0}

When the probe is moved parallel to the plates the potential remains the same: The probe is moving along an equipotential, that is, a plane of equal voltage or potential. The equipotentials are always at right angles to the field lines. Notice that at the edges of the plates where the field is no longer uniform, the equipotentials are still perpendicular to the field lines. The potential changes gradually in a uniform field. In a graph of potential against distance between the plates. The plot shows that the potential gradient is constant and negative. Thus, where r is the distance, the field strength (the force on unit charge) is given by the equation:

Field strength = -(potential gradient)

E = -dV/dr

Although this relationship has been derived for a uniform field, we shall see later that it is true for all fields.

^{Potential graph. Meriney, CC BY-SA 4.0}

NON-UNIFORM FIELDS

In non-uniform fields the potential gradient varies. In the electric field below a thundercloud. Notice how the potential gradient increases around the pointed church spire. The pointed spire distorts the pattern, making the field stronger around the point.

This property is used in a lightning conductor, which is basically a pointed metal rod. One end is earthed and the pointed end extends above the top of the building to which it is attached. A thundercloud often has a concentration of negative charge at its base. This generates an electric field between the base and the ground. Below the thundercloud, the field at the point of the lightning conductor is strong enough to ionize air molecules around the point.

The resulting charged molecules – ions – move in the field: positive ions go towards the cloud base and electrons go to earth through the lightning conductor. This mechanism allows the thundercloud to discharge slowly before enough charge builds up to cause lightning.

Electrostatic paint spraying

Particles of paint are given a positive charge as they leave the nozzle of a spray gun. The object to be painted is earthed so that there is an electric field between the nozzle and the object. The charged paint droplets follow the field lines and are deposited evenly over the surface of the object

PARALLEL PLATE CAPACITOR

So far, I have been looking at the electric field between two parallel plates. This arrangement can in fact store electric charge, which makes it a capacitor. We have seen that the field between the capacitor plates depends upon the p.d across them and the distance between them. Let us now look at other factors that affect the strength of a field.

There is a p.d. across the plates when they are charged. In fact, the charge on the plates is proportional to the p.d. (Q = CV). The quantity of charge on the plates also depends on the area of the plates. If the area of the plates is doubled, the quantity of charge the capacitor can store is doubled. So:

Q α A

The ratio of charge to area (Q/A) is called the charge density (σ). Since the charge density is proportional to p.d., it is proportional to the field strength:

charge density α field strength

σ α E

Q/A α V/d

There is one other factor that affects the field strength. This is the medium between the plates, the dielectric. The dielectric is an insulator, the simplest being a vacuum. Often, the insulating material is air, oil or paper. For a vacuum between the plates:

Q/A = ε₀ V/d

Where ε₀is a constant called the permittivity of free space (or of a vacuum) which links charge density and field strength. Its units are farads per metre and it tells us how good a vacuum is at allowing an electric field to be established. The farad, symbol F, is the unit of capacitance: a 1 farad capacitor charged by a p.d of 1 volt carries a charge of 1 coulomb. In practise, capacitors have much smaller values, ranging between microfarads and picofarads.

The value of ε₀is 8.85 × 10^-12 Fm^-1, or alternatively 8.85 × 10^-12 C² N^-1 m^-2.

Another way of arranging the equation gives expressions for capacitance, Symbol C:

C = Q/V = ε₀ A/d

The second line of the equation gives the capacitance in terms of the physical dimensions of the dielectric medium and the capacitor. In practice, different dielectric materials are used for capacitors, depending on their size.

The equation for the capacitance is slightly modified to account for the dielectric:

C = ε₀ ε_r A/d

Where ε_r is called the relative permittivity of the dielectric used. The plates of a capacitor are covered with a ‘blanket’ of charge. The charge spreads out evenly over most of the plates – only at the edges is the distribution more uneven with more charge concentrated at any ‘sharp’ edges and corners.

^{Capacitors. Eric Schrader, CC BY S.A 2.0}

PRACTICAL CAPACITORS

There are many different types of capacitor available, made from different materials and suitable for different applications. They come in a range of shapes and sizes, in values from picofarads up to farads. Capacitors are used to store charge (and energy), to ‘smooth’ a fluctuating voltage or to allow high frequency signals to ‘bypass’ parts of a circuit. They are essential in circuits designed to respond to certain frequencies. Examples are the ‘tuned circuit’ in radio receivers used to select a signal of a particular frequency, and ‘filters’ which use the fact that the behaviour of a capacitor in a.c. circuits depends on frequency. With accurate capacitors, filters will respond to very narrow bands of signal frequencies, which can be either filtered out or allowed to pass.

Sometimes very accurate capacitors are needed, usually of low value,typically pF or nF. Capacitors of very high value are used, such as in ‘smoothing’, where accuracy is not important. All capacitors lose charge over time, that is, they ‘leak’. Most capacitors have a fixed value, but others vary over a range of values, as in tuned circuits. The table below shows some of the common types of capacitor that are used.

Type	Capacitance range	Maximum voltage	Accuracy	Leakage
Mica	1 pF-10 nF	100-600 V	Good	Very useful at radio frequencies
Ceramic	10 pF-1 uF	50-30 000 V	Poor	cheap, small
Polystyrene	10 pF-2.7 µF	100-600 V	Excellent	High quality, used in accurate filters
polycarbonate	100 pF-30 pF	50-800 V	Excellent	High quality, small
Tantalum	100 nF-500 µF	6-100 V	Poor	High capacitance, polarised
Electrolytic (aluminium)	100 nF-2 F	3-600 V	Horrible	Smoothing power supply filters, polarised

EFFECT OF THE DIELECTRIC

Placing an insulating material between the plates of a capacitor increases the capacitance. That means that more charge can be stored for the same voltage across the plates, and this effect is greater when materials with greater values of ε_r are used.

NON-UNIFORM ELECTRIC FIELD – COULOMB’S LAW

In 1785, Coulomb measured the forces between small charged bodies and summarised his results in the form of an equation that is now known as Coulomb’s law. He discovered that force F depends on the size of the charges Q₁ and Q₂ on the bodies, and on the distance r between them. As the distance increases, the force decreases – in fact, if the distance doubles, the force decreases by a factor of four. That is, the force obeys an inverse square law:

F = k Q₁Q₂/r²

where k is constant.

This equation is almost identical in form to Newton’s law of gravitation:

F = -G m₁m₂/r²

Unlike gravity, which is always attractive, the electric force can be either attractive or repulsive: when the charges are opposite, the force is attractive and defined as negative, like gravity; when the charges are of the same sign, the force is repulsive and defined as positive.

The figure below shows the field around an isolated charge. This is a two-dimensional representation, but the field around the charge is three-dimensional. The field lines spread out radially from the centre. Yet again, this ‘map’ of field lines helps us visualise the field, but it is limited. It is very difficult to show on such a ‘map’ the precise way that the magnitude of the field changes. The equations below help solve this problem.

^{Field of a positive and a negative point charge. Geek3, CC BY-SA 3.0}

FIELD STRENGTH

The strength E of the electric field can be measured by considering the force on a second ‘test’ charge, Q₂, at a distance r from a charge Q₁:

Field strength = force per unit charge

= (k Q₁Q₂/r²) ÷ Q₂

E = k Q₁/r²

That is, the field strength in a non-uniform electric field also obeys the inverse Square law.

EXAMPLE

The electric field strength 50 cm from the centre of a small charged sphere is 18 NC^-1. Calculate the charge on the sphere. (k = 9 × 10⁹Nm² C^-2)

ANSWER: The field strength is:

E = k Q/r²= 18 NC^-1

Rearranging:

Q = Er²/k = 18 × ( 5 × 10^-1)² / 9 × 10⁹

= 18 × 25 × 10^-2/9 × 10⁹

= 5 × 10^-10 C

POTENTIAL IN A RADIAL FIELD

The flame probe used to investigate the potential between parallel plates was described earlier. It can also be used to investigate the way electrical potential varies around an isolated charged body, in this case a sphere. The potential decreases, but it follows a 1/r law

The potential again depends on the charge:

V = k Q₁/r

The constant k can be found by considering an isolated charged sphere. In such a sphere, with radius r, carrying a charge of Q coulombs. We know that:

Charge density = ε₀ × field strength

Q/A = ε₀kQ/r²

Where A = 4πr², the area of the sphere. Therefore we have:

Q/4πr² = ε₀kQ/r²

The charge and radius squared cancel, leaving us with:

k = 1/4πε₀

RELATIONSHIP BETWEEN FIELD AND POTENTIAL

Earlier in this post, I showed that for a uniform field the field strength E was equal to minus the potential gradient:

E = -dV/dr

The same is still true for fields that vary by an inverse square law.

I can show this mathematically:

V = 1/4πε₀ × Q/r
E = -dV/dr
= - Q/4πε₀ × d(1/r)/dr
Therefore:
E = 1/4πε₀ (Q/r²)

This means that the slope of the potential-distance graph gives the value of the field strength at that point.

SUMMARY

By the end of this post and the [previous one] you should know and understand the following:

.The field between charged plates is uniform and the electric field strength is the force per unit charge measured in newtons per coulomb.

.The equations for uniform fields are E = F/Q = V/d and the work done in moving a charge of Q coulombs through a p.d. of V volts is QV joules.

.The kinetic energy of an electron emerging from an electron gun is given by mv² = eV and the electronvolt is the energy gained by a particle carrying a charge of 1.6 × 10^-19 coulombs when it has been accelerated through a p.d. of 1 V.

.The potential at a point in an electric ficld is the work done per coulomb in moving a unit charge from zero or earth potential to that point. Equipotentials can be thought of as surfaces in space of equal potential which cut electric field lines at right angles.

.Field strength E is equal to (minus) the potential gradient, -dV/dr. Charge density = permittivity × field strength of free space, and capacitance C = ε₀A/d and the force between two charges is given by Coulomb’s law, F = k Q₁Q₂/r² and field strength E = kQ/r², where k = 1/4πε₀ and the potential around a charge Q is given by V = kQ/r.