Topic 9 - Electric Fields

Coulomb’s Law

There are many parallels between gravity fields and electric fields.  The electric field is the region of force around a point charge.  In 1784 Coulomb showed that the force between two point charges was proportional to each charge and inversely proportional to the square of the distance between them.  This reminds us of Newton’s Law of Gravitation. 

Coulomb’s Law can be summarised in code as:

We can rewrite this proportionality as:

The constant k has a value of 8.99 ´ 10⁹ Nm² C^-2 for two charges in a vacuum.  The constant is not normally written as k, but instead:

 [e₀ = 8.85 ´ 10^-12 C² N^-1 m^-2 or F/m]

This term e₀ (‘epsilon nought’ - e is a Greek letter ‘e’) is called the permittivity of free space.  We can write the units for k as m/F.  The equation is written as:

Since you have to know this formula, the easiest way is:

We can approximate the value of k to 9.0 ´ 10⁹ m/F.  Its large value tells us why electrical forces are very strong compared to gravity forces, even when the charges are quite small.  This helps us to explain why solid materials can exist at all.

The force F is positive when the charges are both positive or both negative; this indicates that the force is repulsive, which is borne out by our elementary observations of electrostatics.  If one charge is positive and the other is negative, we can easily see that the force F is negative, therefore attractive.

Electric Field Strength

A gravity field is where a force is exerted on a mass; an electric field is a region where a force is exerted on a charge.

The electric field strength is defined as force per unit charge.  In equation form this is represented as:

E = F/Q    

[E – electric field strength; F – force; Q – charge]

E is a vector quantity and its units are newtons per coulomb, N/C. 

Since the electric field strength is defined as force per unit charge, we can write:

This equation is true for all point charges, which can be considered to have a radial field.  A spherical charge can also be considered to have a radial field provided the distance from the charge to the centre is much greater than the radius of the sphere.

We can see that the graph shows a classic inverse square law.

Where two charged plates are close together, the radial fields of the charges combine to make a uniform field.

Notice that the field bulges at the ends; generally we ignore this.

In this case we can show that the electric field strength is given by a simpler relationship:

E = V/d          

[E – electric field strength; V – voltage; d distance in m]

Strange as it may appear, the units of volts per metre are entirely consistent with newtons per coulomb.

The direction of the field is the direction of the force.  The force in this case is the force exerted by the electric field on a hydrogen ion (proton).

If we are looking at the electric field between two point charges, we need to take both into account both the charges.

Let us look at a more complicated example:

Electric Potential, V

When we looked at gravity fields, we saw that gravitational potential energy is the energy needed to bring an object from infinity to a certain point.  We also saw that gravitational potential was the energy per unit mass.  There are two similar quantities in electric fields, potential energy and electric potential.

Suppose we moved a charge form one point to another in an electric field.  We would have to do a job of work against the field (or get work out if it were with the field), so there is an energy change.  We can show this as a graph:

The area under the graph represents the energy transformed.  Using the mathematical trick of calculus, we can show that the energy is given by the equation:

We can define the electrical potential as the energy per unit charge.  It has the code Ve and is given by the equation:

The units are Joules per Coulomb (JC-1) or Volts (V).  Indeed the volt is defined as the energy transformed when unit charge is moved between two points.  We can show the relationship between the electric field and potential in this graph.  Potential is the area under the graph:

All points within an electric field that have the same potential are called equipotentials, rather like contours on a map.  In a radial field, the equipotentials are concentric circles.  In a uniform field, they are parallel lines equally spaced.

Worked Example

Trajectories of Particles in Electric Fields

The key points to remember are:

·        The electric field goes from positive to negative.

·        The electron carries a charge of –1.6 ´ 10^-19 C.

·        The electron is attracted to the positive side, so goes against the field.

·        The proton carries a charge of +1.6 ´ 10^-19 C.

·        The proton is repelled by the positive side, so moves in the direction of the field.

·        Other charged particles have whole number multiples of 1.6 ´ 10^-19 C.

Electrons can be accelerated by an electric field.  This happens in any cathode ray tube, e.g. a TV set, or a VDU.  Electrons are “boiled off” a red hot cathode in a process called thermionic emission.  They are then attracted by a positive electrode, the anode, at a high voltage.  This makes them move forward at a high speed.  The electrons move parallel to the direction of the field.  Most hit the anode, but some fly through a hole at the front, to hit a screen at the far end of the tube.  The screen is covered in phosphor, and the energy the electrons have is converted to light (and some X-rays).  Because this arrangement spits out electrons, it is called an electron gun.

The electrons are repelled by the cathode and accelerated by the anode.  Each electron is given energy, which can be found using energy = charge × voltage, E = QV.  The energy is entirely kinetic energy, so we can say that:

            QV = Ek = ½ mv²

            Þ v² = 2QV

                          m

There is no reason why positive charges cannot be accelerated in the same way.  Indeed in a mass spectrometer, positive ions are accelerated by an electric field and bent by a magnetic field to hit a detector.  Particle accelerators work in the same way.

Let us consider how an electron behaves as it enters a uniform electric field at right angles:

• We know that the electric field strength is given by E = V/d

• From our definition of electric field we can say that the force on the electron is given by   F = EQ.

• Therefore the force is given by combining these two equations:

• We can apply Newton II to help us to work out the acceleration.  If we wanted to know the upwards velocity, we would work out the time interval, and then use a = Dv/Dt to calculate the upwards velocity.  Since the electrons are in a vacuum, the horizontal velocity is not affected.

• Once we know the upwards velocity, then we can do a vector addition to tell us the resultant velocity.

Comparing Electric and Gravitational Fields

There are many analogies that can be drawn between electric fields and gravity fields.  Theoretical physicists would go as far as saying that the two are possibly different manifestations of the same thing.  Let us compare the two:

The gravitational attraction between particles in an atom is so small as to be negligible.  The nucleus and its electrons are held together entirely by electrostatic forces, and these are involved in chemical reactions.

Gravity forces hold planets together and hold them in their orbits.  Electrostatic forces over the interplanetary distances can be ignored.

Summary Force between charges:                                                    k =   1  = 9 ´ 109 m/F      4pe0   Electric Field Strength: Force per unit charge                                                      E = F/Q   For a uniform field:                        E = V/d   Electric Potential: Energy per unit charge                                                     Potential Energy: Energy needed to bring a charge from infinity to a second charge.                                                     Ep = k Q1Q2                                                                 r