**ELECTRIC FIELDS**

In the following set-up, two strips of polythene have been rubbed with wool, and when bought close together, they repel each other:

The polythene has become 'electrostatically charged' by friction with the wool.

If a strip of Perspex is rubbed with wool, and then bought close to the suspended polythene strip, they attract each other.

This indicates that there are *two types of charge*, and
we call these *positive and negative*. The polythene strips are taken as
having a negative charge, and the Perspex a positive charge.

Observations like those above indicate that:

- negative repels negative
- positive repels positive
- negative and positive attract each other

*Like charges repel and unlike charges attract*

- An atom has a nucleus which contains positively charged protons and uncharged (neutral) neutrons
- In orbit around the nucleus are negatively charged electrons
- The charge of protons and electrons are the same size but opposite in sign
- Uncharged materials contain atoms with equal numbers of protons and electrons, so their charges cancel each other
- Materials can become electrically charged by rubbing - they gain or lose electrons becoming negatively or positively charged

*The force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of their separation*

The above represents the situation when the forces are attractive. No subscripts are needed on the Fs, since the force on each charge is the same in magnitude, but they are opposite in direction (recall Newton’s third law). According to Coulomb's law:

Note: For two charged spheres, r is the distance
between their centres.

**Permittivity**

Notice how similar the above is to the gravitational case. However, one difference is that in the gravitational case, the force between two masses is always attractive, whereas in the electric case the force can be attractive or repulsive, depending on the signs of the charges.

Also, in the gravitational case, the constant of
proportionality is the universal gravitational constant G, and its value does
not depend upon the nature of the medium between two masses. However, in the
electric case the force between two charges *does* depends upon what is
between them.

To take the medium into account we write Coulomb’s law as:

A more commonly used, and equivalent unit, is
‘*farads per metre’* (F m^{-1})

There is no doubt that the forces between charges are real, since we can observe the effect of such forces, as in the simple experiment with charged strips above. However, there is no apparent physical contact between the strips, and we cannot see what produces the forces. To ‘explain’ such ‘action at a distance’, we assume that an electric charge is surrounded by an invisible 'electric field', and that it is the interaction between fields that produces the observed effects.

An electric field is any region where an electric force may be experienced. We represent such fields by lines with arrows on them.

The direction of the field at a point, represented by an arrow, is defined as the direction of the force on a positive charge at that point. Thus, arrows point away from a positive charge and towards a negative charge.

**ELECTRIC FIELD
STRENGTH (OR INTENSITY) **

**Definition**

*The electric field strength at a point equals the force per unit positive charge at that point*

From the definition, the unit of E = unit of (F/q) = N C

A more commonly used, but equivalent, unit of E is volts per metre (V m

**E due to a point charge**

Consider a small positive test charge q placed at P, a distance r from a point charge +Q:

By Coulomb’s law, the force F on q is:

In the diagram, the point charge is represented as being
positive, but the above equation gives E for both a positive or negative point
charge Q.

**E due to a charged spherical conductor**

So far as external effects are concerned, we can treat a spherical conductor, having charge Q distributed uniformly over its surface, as if all the charge were at its centre.

Thus, if its radius is a, the field strength E at a distance r from its centre (r ³ a):

**Definition:**

**Definition**

*The electric potential at a point in an electric field is numerically equal to the work done in moving a unit positive charge to that point from infinity, where the potential is defined as zero*

Potential is a scalar quantity, and the potential at a point due to a number of charges is the algebraic sum of the potentials at that point due to each charge.

**Potential due to an isolated point charge**

We can show that the electric potential V

And also that:

(Hence, the unit V/m for electric field strength, E.)

It can be shown that:

i) The potential V at a distance r from the centre is given by:

Where, r ³ a, i.e. on the surface and outside the sphere.

ii) Inside the sphere the potential is the same at all points and the same as on the surface. Thus,

*The p.d. between two points in an electric field is numerically equal to the work done in moving a unit positive charge from one point to the other*

Thus, the work W done in moving a charge Q through a p.d. V is given by:

Note: The work done in taking a charge around a closed loop is zero - since it arrives back at its original point, and so the p.d. between its start and end points is zero.**Parallel plates**

A battery is applied to two flat, parallel metal plates and produces an electric field as represented above.

Near the edges the field in not uniform, and so is not given accurately by this equation.

**Equipotentials**

In an isolated conductor there can be no differences in potential, since these would set up potential gradients (º electric fields), and charges would redistribute themselves until they had destroyed the field.

*Any surface or volume over which the potential is constant
is called an equipotential*. The surface or volume may be that of a material
body or a surface or a volume in space.

Since the change in potential between any two points in an equipotential surface is zero, there is no potential gradient, so there is no electric field. Electric field lines are therefore at right angles to equipotential surfaces, since the field lines then do not have components in the surface.

a) All spheres centred on a point charge are equipotential surfaces:

b) The potential gradient between two parallel metal plates is constant (apart from at the edges) and all planes parallel to and between the plates are equipotential surfaces:

Electric field strength, E = 3V/3cm = 1 V/cm.

The above represents the basic structure of a capacitor.

**CAPACITORS** **BASIC CHARACTERISTICS**

A capacitor is a device that can store electric charge. It is basically a very simple device consisting of two metal sheets, separated by an insulating material. Often, in practical capacitors, the sheets are rolled up, so the capacitor becomes cylindrical, and is similar to a roly-poly pudding in cross-section.

The above represents a capacitor made of two metal plates, with a battery applied across them. There is a current while the capacitor is ‘charging up’ - electrons flow from one plate to the other. When charging is complete, the p.d. across the capacitor equals that of the battery.

When charged up, an electric field exists between the plates. The direction of the field is defined as that of the force on a positive charge placed between the plates. If charge q were between the plates and experienced a force F, then the magnitude of the electric field, denoted by E.

From this we get the more
commonly used, but equivalent, unit for E of volts per metre
(Vm^{-1}).

If the voltage across a capacitor is too great, the insulator breaks down, and becomes a conductor. This can make the capacitor get hot or even explode. The working voltage of a capacitor is normally written on it, indicating the maximum voltage that can be safely applied to it.

**Capacitance**

The capacitance C of a capacitor is a measure of its ability to store charge. By definition:

From the definition, the unit of capacitance is C/V or CV^{-1 }('coulombs per volt'), which we
call the farad, F: i.e. 1F = 1CV^{-1}.

**Energy stored in a capacitor**

A graph of p.d. versus charge is a straight line through the origin:

**Example**

- find the charge on the capacitor
- find the energy stored using each of the above equations

**Capacitors in parallel**

Key facts:

- There is the same p.d. V across each capacitor
- The total charge stored, Q = Q
_{1}+ Q_{2}+ Q_{3}

C may be referred to as the equivalent or effective or combined capacitance.

By fact (1) we do not need a subscript on V since
the p.d. is the same for each.

And, by fact (2):

Notice that this expression is similar to that for resistors *in
series*.

**Capacitors in series**

Key facts:

- There is the same charge on each capacitor
- V = V
_{1}+ V_{2}+ V_{3}

By fact (1) we do not need a subscript on Q since
the charge is the same for each.

And, by fact (2):

Notice that this expression is similar to that for resistors *in
parallel*.

**Example**

Find the single capacitor equivalent to:

**Example**

Find the single capacitor equivalent to:

Find the single capacitor equivalent to:

For the capacitors in series,

Notice that V

**Joining two capacitors**

The two capacitors are initially separate. When joined (the dotted lines):

- The total capacitance, C = C
_{1}+ C_{2} - The capacitors acquire the same p.d.
- The total charge remains constant

In (a), capacitor C_{1} has been charged by a 60V
supply.

In (b), C_{1} has been joined across an
uncharged capacitor C_{2}.

We see that the final amount of stored energy is less than the initial amount. When the capacitors are joined, current has to flow to achieve the redistribution of charge, and this generates some heat loss.

**DISCHARGING A CAPACITOR THROUGH A RESISTOR**

With S in position

When S is moved to position 2, the capacitor will start to discharge through R:

**a) Charge**

The charge starts at its maximum value and then falls to zero:

The 'decay' is referred to as being 'exponential', the variation of Q with time obeying the equation:

Where 'e' is a number with the value 2.718 ........... It is the base of 'natural logarithms'.

The magnitude of the power that e is raised to, (t/CR), has no
unit. So, since time t is in seconds, then CR must also be in seconds, in order
that the units on top and bottom cancel. Thus, CR has the unit of time, and is
called the ‘*time constant*’ of the circuit - it determines the rate at
which Q approaches zero:

Theoretically, the above curves never reach the time axis, since time t has to equal ‘infinity’ for Q to equal zero. Hence, to compare different CR circuits, we introduce the idea of 'half life' of a CR circuit. The term is analogous to the same term used in radioactive decay.

*The half-life of a discharging capacitor is the time for the charge on the capacitor fall to half its initial value*

To follow the derivation below you may have to read up about
'logarithms' - your syllabus may not require you to know the derivation - but
the final relationship, between CR and T_{1/2}, *is* important:

Thus, *the capacitor is half discharged when the time reaches
69% of the time constant*.

While the capacitor is discharging:

- the current in the circuit starts at a maximum value
I
_{0}, and falls to zero - the p.d. across the capacitor starts at a maximum value
V
_{0}, and falls to zero

- the charge to fall to Q
_{0}/2 - the current to fall to I
_{0}/2 - the p.d. to fall to V
_{0}/2

**b) Current**

Current equals rate of flow of charge. Thus, the slope (or gradient) of the Q-t graph at any time equals the current at that time:

The equation for this comes directly from that for Q, since V = Q/C

**CHARGING A CAPACITOR THROUGH A RESISTOR **

When the switch S is closed, the charge on the capacitor rises
from zero to its maximum value of Q_{0} = CV_{0}. The
variation of charge Q with time t has the form:

The charge Q on the capacitor at time t seconds after the switch is closed is given by the equation:

We again use the idea of half-life. Its meaning in this case is indicated by the above graph for charge and the graphs below for current and voltage. The value of half-life is the same in each case, and found from the same equation as before:

We can show that:

**c) Potential difference**

The equation for this comes directly from that for Q, since V = Q/C

A Level Physics - Copyright © A
C Haynes 1999 & 2004