Lecture 2 Electric fields

In this lecture the following are introduced:

• the idea of electric fields,

• field strength, and

• potential difference in a field.

Forces that operate through physical contact are explained by the concept that two objects are not able to occupy the same space at the same time.

Forces that operate without physical contact require a different explanation.

Properties such as mass and electric charge actually change the structure of the space around them.

People normally think of the structure of space around them as "straight and right-angled" i.e. like in a three dimensional Cartesian co-ordinate system.

This is called Euclidean space (and represented in two-dimensions as shown below).

However the space around isolated masses and charges is not Euclidean:

it is spherically curved.

This means that an electric field of influence extends from an isolated electric charge, radially out into the space around it, and other charges are influenced by it.

The words of "electric field of influence" are shortened to "electric field".

An electric field can be mapped by the force on a (very small positive) "test" charge placed in the field. The test charge has to be small, so that it does not appreciably alter the field it is mapping.

This mapping produces "lines of force" where the clustering of lines indicates a strong field and the tangent gives the direction of the force.

Electric field mapping

Electric field lines start on positive charges and end on negative charges.

The electric field around an isolated positive charge is radially outwards (spherical symmetry)

The electric field around an isolated negative charge is radially inwards (spherical symmetry).

The electric field around two isolated positive charges

The electric field around two isolated negative charges

The electric field around a positive charge and an equal but negative charge. (This arrangement is called a dipole.)

Electric Field Strength

To compare things exactly, we need to define a quantitive measure of the strength of an electric field.

The Electric Field Strength (or Electric Field Intensity), E, at a point in space is measured by the force, F, acting on a small positive test charge, q placed at that point.

The unit for Electric Field Strength is Newtons per Coulomb, symbolically as N.C^-1.

Ideally, (and impractically) the test charge will be vanishingly small, so more correctly:

Note that often Electric Field Strength/Intensity is just referred to as Electric Field.

Electric Field Strength for an isolated point charge

The Electric Field Strength at a distance, r, from an isolated point charge of size, q₁ can tested with the small charge, q, and deduced from Coulomb's law.

This expression can be written as:

This indicates that the product of Field Strength and Surface Area around the charge is proportional to what causes the field (the charge enclosed).

It says that the bigger the area: the smaller the field strength, because the influence has to spread out more "thinly".

Electric Field strength is a vector so directions must be taken into account.

Example

Two isolated charges of +3mC and -5 mC are fixed 0.3 m apart in vacuum.

Find the electric field strength at the midpoint, P, between them.

Example

For the fixed isolated charges shown, find

(a) the electric field strength and

(b) the electric force ,

on a +4mC charge placed at the centre of the equilateral triangle shown below.

The first task is to find the electric field strength at the centre of the triangle.

To find the centre of the triangle, drop perpendiculars from the vertices.

The distance, r, from each charge to the centre, is given by:

Take the origin as the centre of the +3mC charge.

This gives a resultant of:

Using Pythagoras' theorem:

The electric force F on a charge q in an electric field E is given from the definition of field strength as:

F = q E

Here the bold type indicates the vector nature of Force and Field.

A positive charge has a force in the same direction as the electric field.

A negative charge has a force in the opposite direction to the electric field.

From above:

Electric Field Strength in the space between two oppositely charged parallel plates

Mapping the Electric field shows that (apart from some fringing at the edges) there is a uniform field between the plates.

Using the idea above:

Dividing by the area:

Here σ is the surface charge density or charge per unit area.

The field between the plates is a uniform field because the lines do not diverge or converge so it has the same value everywhere.

(Like the value of g, the acceleration of gravity, is the same in a room).

Example

Two equally sized circular parallel plates are separated by 1mm of air and have radii of 0.03 m.

They store a charge of 7mC.

Find the electric force on a -5mC charge placed anywhere between the plates.

The Electric Field Strength between the plates is directly away from the positive plate and directly towards the negative plate:

The electric force on the charge is given by:

Electric potential difference

Electric Potential Difference is different from Electric Potential Energy Difference.

The unit of Electric Potential is Joule/Coulomb, which is called the Volt.

The definition shows that an Electric Potential Difference is the effect produced by an Electric Field between two points.

Since potential is a scalar, we do not need to take the directions into account.

It is also easier to measure potential difference than use test charges to find electric field strength.

Note: To get back the electric field strength from the potential difference

Here the dr introduces a direction for the field strength.

from this, E also has the unit of V.m^-1.

When the electric field is not uniform (ie it changes with distance), then you have to integrate to find the potential difference.

Electric Potential at a distance from isolated positive charge

Since potential difference is a difference, it is relative to a separation, so there is no absolute zero for it.

However, it is often convenient to set zero potential at an infinite distance from a charge.

The potential at a distance, r , from an isolated charge, q, is then the potential difference between infinity and that distance.

This is calculated from the potential energy change in moving a charge q₁ from infinity to a distance r from the charge q.

This has spherical symmetry so all points on the surface of a sphere of radius r around q have the same potential.

Equipotential Surfaces

An equipotential surface is one on which the points are all at the same electrical potential and there is no net electrical work done in moving from one point to another.

Equipotentials for isolated charges

For isolated charges the equipotential surfaces are spheres centred on the charge.

Note that equipotential surfaces are always at right angles to lines of force.

Equipotentials for two like charges

Equipotentials for a dipole

Example

Find the electric potential at a distance of 2.56nm from a Carbon atom nucleus that has a positive charge equal to 6 electrons.

Example

A charge of +3mC and a charge of -7mC are 0.4 m apart in air.

Find the distance from the +3mC towards the -7mC charge where the electric potential is zero

Since potential is a scalar quantity, we do not need to take the directions into account.

It is also easier to measure potential difference than use test charges to find electric field strength.

Electric Potential Difference between charged parallel plates

Since the field is uniform, (E is constant), two points in the field separated by a distance d along the direction of the field lines have a potential difference:

V=E⋅d

Example

Two parallel conducting plates have an area of 10^-2 m² and carry a charge of 8nC. If the separation (in air) of the plates is 3mm, find the potential difference between the plates.

Summarising:

Electric charges have fields around them.

Electric field lines start on positive charges and end on negative charges.

Electric Field Strength is in N.C^-1

.

E is a vector and V is a scalar.

Equipotentials are surfaces of equal potential at right angles to field lines.

For an isolated charge in air:

For charged parallel plates in air:

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