Notice how similar Coulomb's Law is to the form of Newton's Universal Law of Gravitation between two point-like particles:

where m1m1 and m2m2 are the masses of the two particles, rr is the distance between them, and GG is the gravitational constant.

Both laws represent the force exerted by particles (masses or charges) on each other that interact by means of a field.

The electric field maps depend very much on the charge or charges that the map is being made for. We will start off with the simplest possible case. Take a single positive charge with no other charges around it. First, we will look at what effects it would have on a test charge at a number of points.

Electric field lines, like the magnetic field lines that were studied in Grade 10, are a way of representing the electric field at a point.

At each point we calculate the force on a test charge, qq, and represent this force by a vector.

Figure 7

We can see that at every point the positive test charge, qq, would experience a force pushing it away from the charge, QQ. This is because both charges are positive and so they repel. Also notice that at points further away the vectors are shorter. That is because the force is smaller if you are further away.

Negative charge acting on a test charge

If the charge, Q, were negative we would have the following result.

Figure 8

Notice that it is almost identical to the positive charge case. This is important – the arrows are the same length because the magnitude of the charge is the same and so is the magnitude of the test charge. Thus the magnitude (size) of the force is the same. The arrows point in the opposite direction because the charges now have opposite sign and so the positive test charge is attracted to the charge. Now, to make things simpler, we draw continuous lines showing the path that the test charge would travel. This means we don't have to work out the magnitude of the force at many different points.

Electric field map due to a positive charge

Figure 9

Some important points to remember about electric fields:

• There is an electric field at every point in space surrounding a charge.
• Field lines are merely a representation – they are not real. When we draw them, we just pick convenient places to indicate the field in space.
• Field lines usually start at a right-angle (90^o) to the charged object causing the field.
• Field lines never cross.

We will now look at the field of a positive charge and a negative charge placed next to each other. The net resulting field would be the addition of the fields from each of the charges. To start off with let us sketch the field maps for each of the charges separately.

Electric field of a negative and a positive charge in isolation

Figure 10

Notice that a test charge starting off directly between the two would be pushed away from the positive charge and pulled towards the negative charge in a straight line. The path it would follow would be a straight line between the charges.

Figure 11

Now let's consider a test charge starting off a bit higher than directly between the charges. If it starts closer to the positive charge the force it feels from the positive charge is greater, but the negative charge also attracts it, so it would experience a force away from the positive charge with a tiny force attracting it towards the negative charge. If it were a bit further from the positive charge the force from the negative and positive charges change and in fact they would be equal in magnitude if the forces were at equal distances from the charges. After that point the negative charge starts to exert a stronger force on the test charge. This means that the test charge would move towards the negative charge with only a small force away from the positive charge.

Figure 12

Now we can fill in the other lines quite easily using the same ideas. The resulting field map is:

Figure 13

Two like charges : both positive

For the case of two positive charges things look a little different. We can't just turn the arrows around the way we did before. In this case the test charge is repelled by both charges. This tells us that a test charge will never cross half way because the force of repulsion from both charges will be equal in magnitude.

Figure 14

The field directly between the charges cancels out in the middle. The force has equal magnitude and opposite direction. Interesting things happen when we look at test charges that are not on a line directly between the two.

Figure 15

We know that a charge the same distance below the middle will experience a force along a reflected line, because the problem is symmetric (i.e. if we flipped vertically it would look the same). This is also true in the horizontal direction. So we use this fact to easily draw in the next four lines.

Figure 16

Working through a number of possible starting points for the test charge we can show the electric field map to be:

Figure 17

Two like charges : both negative

We can use the fact that the direction of the force is reversed for a test charge if you change the sign of the charge that is influencing it. If we change to the case where both charges are negative we get the following result:

Figure 18

One very important example of electric fields which is used extensively is the electric field between two charged parallel plates. In this situation the electric field is constant. This is used for many practical purposes and later we will explain how Millikan used it to measure the charge on the electron.

Field map for oppositely charged parallel plates

Figure 19

This means that the force that a test charge would feel at any point between the plates would be identical in magnitude and direction. The fields on the edges exhibit fringe effects, i.e. they bulge outwards. This is because a test charge placed here would feel the effects of charges only on one side (either left or right depending on which side it is placed). Test charges placed in the middle experience the effects of charges on both sides so they balance the components in the horizontal direction. This is clearly not the case on the edges.

Strength of an electric field

When we started making field maps we drew arrows to indicate the strength of the field and the direction. When we moved to lines you might have asked “Did we forget about the field strength?”. We did not. Consider the case for a single positive charge again:

Figure 20

Notice that as you move further away from the charge the field lines become more spread out. In field map diagrams, the closer together field lines are, the stronger the field. Therefore, the electric field is stronger closer to the charge (the electric field lines are closer together) and weaker further from the charge (the electric field lines are further apart).

The magnitude of the electric field at a point as the force per unit charge. Therefore,

E = F q E = F q

(18)

E and F are vectors. From this we see that the force on a charge qq is simply:

F = E · q F = E · q

(19)

The force between two electric charges is given by:

F = k Q q r 2 . F = k Q q r 2 .

(20)

(if we make the one charge QQ and the other qq.) Therefore, the electric field can be written as:

E = k Q r 2 E = k Q r 2

(21)

The electric field is the force per unit of charge and hence has units of newtons per coulomb.

As with Coulomb's law calculations, do not substitute the sign of the charge into the equation for electric field. Instead, choose a positive direction, and then either add or subtract the contribution to the electric field due to each charge depending upon whether it points in the positive or negative direction, respectively.

Figure 21

Khan academy video on electrostatics - 2

Figure 22

Phet simulation for Electric Fields

Exercise 5: Electric field 1

Calculate the electric field strength 30 cm 30 cm from a 5 nC 5 nC charge.

Figure 23

Solution

1. Step 1. Determine what is required :

   We need to calculate the electric field a distance from a given charge.

2. Step 2. Determine what is given :

   We are given the magnitude of the charge and the distance from the charge.

3. Step 3. Determine how to approach the problem :

   We will use the equation:

   E = k Q r 2 . E = k Q r 2 .

   (22)
4. Step 4. Solve the problem :
   E = k Q r 2 = ( 8 . 99 × 10 9 ) ( 5 × 10 - 9 ) ( 0 , 3 ) 2 = 4 , 99 × 10 2 N . C - 1 E = k Q r 2 = ( 8 . 99 × 10 9 ) ( 5 × 10 - 9 ) ( 0 , 3 ) 2 = 4 , 99 × 10 2 N . C - 1

   (23)

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Exercise 6: Electric field 2

Two charges of Q1Q1 = +3 nC +3 nC and Q2Q2 = -4 nC -4 nC are separated by a distance of 50 cm 50 cm . What is the electric field strength at a point that is 20 cm 20 cm from Q1Q1 and 50 cm 50 cm from Q2Q2? The point lies beween Q1Q1 and Q2Q2.

Figure 24

Solution

1. Step 1. Determine what is required :

   We need to calculate the electric field a distance from two given charges.

2. Step 2. Determine what is given :

   We are given the magnitude of the charges and the distances from the charges.

3. Step 3. Determine how to approach the problem :

   We will use the equation:

   E = k Q r 2 . E = k Q r 2 .

   (24)

   We need to work out the electric field for each charge separately and then add them to get the resultant field.

4. Step 4. Solve the problem :

   We first solve for Q1Q1:

   E = k Q r 2 = ( 8 . 99 × 10 9 ) ( 3 × 10 - 9 ) ( 0 , 2 ) 2 = 6 , 74 × 10 2 N . C - 1 E = k Q r 2 = ( 8 . 99 × 10 9 ) ( 3 × 10 - 9 ) ( 0 , 2 ) 2 = 6 , 74 × 10 2 N . C - 1

   (25)

   Then for Q2Q2:

   E = k Q r 2 = ( 8 . 99 × 10 9 ) ( 4 × 10 - 9 ) ( 0 , 3 ) 2 = 2 , 70 × 10 2 N . C - 1 E = k Q r 2 = ( 8 . 99 × 10 9 ) ( 4 × 10 - 9 ) ( 0 , 3 ) 2 = 2 , 70 × 10 2 N . C - 1

   (26)

   We need to add the two electric fields beacuse both are in the same direction. The field is away from Q1Q1 and towards Q2Q2. Therefore,

   Etotal=6,74×102+2,70×102=9,44×102N.C-1Etotal=6,74×102+2,70×102=9,44×102N.C-1

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The electrical potential at a point is the electrical potential energy per unit charge, i.e. the potential energy a positive test charge would have if it were placed at that point.

Consider a positive test charge +Q+Q placed at A in the electric field of another positive point charge.

Figure 26

The test charge moves towards B under the influence of the electric field of the other charge. In the process the test charge loses electrical potential energy and gains kinetic energy. Thus, at A, the test charge has more potential energy than at B – A is said to have a higher electrical potential than B.

The potential energy of a charge at a point in a field is defined as the work required to move that charge from infinity to that point.

Definition 3: Potential difference

The potential difference between two points in an electric field is defined as the work required to move a unit positive test charge from the point of lower potential to that of higher potential.

If an amount of work WW is required to move a charge QQ from one point to another, then the potential difference between the two points is given by,

From this equation we can define the volt.

Definition 4: The Volt

One volt is the potential difference between two points in an electric field if one joule of work is done in moving one coulomb of charge from the one point to the other.

Lightning is an atmospheric discharge of electricity, usually, but not always, during a rain storm. An understanding of lightning is important for power transmission lines as engineers need to know about lightning in order to adequately protect lines and equipment.

Estimating distance of a lightning strike. The flash of a lightning strike and resulting thunder occur at roughly the same time. But light travels at 300 000 kilometres in a second, almost a million times the speed of sound. Sound travels at the slower speed of 330 m/s in the same time, so the flash of lightning is seen before thunder is heard. By counting the seconds between the flash and the thunder and dividing by 3, you can estimate your distance from the strike and initially the actual storm cell (in kilometres).

A parallel plate capacitor is a device that consists of two oppositely charged conducting plates separated by a small distance, which stores charge. When voltage is applied to the capacitor, usually by connecting it to an energy source (e.g. a battery) in a circuit, electric charge of equal magnitude, but opposite polarity, builds up on each plate.

Figure 28: A capacitor (C) connected in series with a resistor (R) and an energy source (E).

Definition 5: Capacitance

Capacitance is the charge stored per volt and is measured in Farads (F).

Mathematically, capacitance is the ratio of the charge on a single plate to the voltage across the plates of the capacitor:

Capacitance is measured in Farads (F). Since capacitance is defined as C=QVC=QV, the units are in terms of charge over potential difference. The unit of charge is the coulomb and the unit of the potential difference is the volt. One farad is therefore the capacitance if one coulomb of charge was stored on a capacitor for every volt applied.

1 C of charge is a very large amount of charge. So, for a small amount of voltage applied, a 1 F capacitor can store a enormous amount of charge. Therefore, capacitors are often denoted in terms of microfarads (1×10-61×10-6), nanofarads (1×10-91×10-9), or picofarads (1×10-121×10-12).

The electric field between the plates of a capacitor is affected by the substance between them. The substance between the plates is called a dielectric. Common substances used as dielectrics are mica, perspex, air, paper and glass.

When a dielectric is inserted between the plates of a parallel plate capacitor the dielectric becomes polarised so an electric field is induced in the dielectric that opposes the field between the plates. When the two electric fields are superposed, the new field between the plates becomes smaller. Thus the voltage between the plates decreases so the capacitance increases.

In every capacitor, the dielectric stops the charge on one plate from travelling to the other plate. However, each capacitor is different in how much charge it allows to build up on the electrodes per voltage applied. When scientists started studying capacitors they discovered the property that the voltage applied to the capacitor was proportional to the maximum charge that would accumulate on the electrodes. The constant that made this relation into an equation was called the capacitance, C. The capacitance was different for different capacitors. But, it stayed constant no matter how much voltage was applied. So, it predicts how much charge will be stored on a capacitor when different voltages are applied.

The capacitance of a capacitor is proportional to the surface area of the conducting plate and inversely proportional to the distance between the plates. It also depends on the dielectric between the plates. We say that it is proportional to the permittivity of the dielectric. The dielectric is the non-conducting substance that separates the plates. As mentioned before, dielectrics can be air, paper, mica, perspex or glass.

The capacitance of a parallel-plate capacitor is given by:

where ϵ0ϵ0 is, in this case, the permittivity of air, AA is the area of the plates and dd is the distance between the plates.

The electric field strength between the plates of a capacitor can be calculated using the formula:

E = V d E = V d

where EE is the electric field in J.C-1J.C-1, VV is the potential difference in volts (VV) and dd is the distance between the plates in metres (mm).

When a capacitor is connected in a DC circuit, current will flow until the capacitor is fully charged. After that, no further current will flow. If the charged capacitor is connected to another circuit with no source of emf in it, the capacitor will discharge through the circuit, creating a potential difference for a short time. This is useful, for example, in a camera flash.

Initially, the electrodes have no net charge. A voltage source is applied to charge a capacitor. The voltage source creates an electric field, causing the electrons to move. The charges move around the circuit stopping at the left electrode. Here they are unable to travel across the dielectric, since electrons cannot travel through an insulator. The charge begins to accumulate, and an electric field forms pointing from the left electrode to the right electrode. This is the opposite direction of the electric field created by the voltage source. When this electric field is equal to the electric field created by the voltage source, the electrons stop moving. The capacitor is then fully charged, with a positive charge on the left electrode and a negative charge on the right electrode.

If the voltage is removed, the capacitor will discharge. The electrons begin to move because in the absence of the voltage source, there is now a net electric field. This field is due to the imbalance of charge on the electrodes–the field across the dielectric. Just as the electrons flowed to the positive electrode when the capacitor was being charged, during discharge, the electrons flow to negative electrode. The charges cancel, and there is no longer an electric field across the dielectric.

Capacitors are used in many different types of circuitry. In car speakers, capacitors are often used to aid the power supply when the speakers require more power than the car battery can provide. Capacitors are also used in processing electronic signals in circuits, such as smoothing voltage spikes due to inconsistent voltage sources. This is important for protecting sensitive electronic components in a circuit.