Field and Potential

Field and Potential

If we know the acceleration of an object, and its starting conditions, then we can integrate to find the velocity, and integrate again to find the position. In this way, we can calculate the whole history of the object. But to know the acceleration, we have to divide the force that acts by the mass, so it becomes important to be able to calculate the forces acting.

This introduction doesn't tackle much in the way of theory - it merely attempts to show that there is a pattern, and that the thirteen equations collected in the final table are really only four basic equations dressed up in differing liveries.

Here is a list of common types of force and the equations for calculating them. First, four familiar ones

Elastic	F = kx	k = stiffness (depends, in turn, on the Young modulus and the dimensions of the specimen)
Pressure	F = pA	A = area (this carries the vector information)
Friction	F_max = μN	μ = coefficient of friction; N = normal contact force. Frictional forces only come into existence when required.
Drag	F = ρAv² F = 6πηrv	A = frontal area. This is for turbulent flow. r = radius of system; η = coefficient of viscosity. This is for laminar flow.

In all of these cases there is a medium of some kind, in contact with the object being acted on. In the next four examples, there is nothing connecting the objects exerting a force on each other, and we have the distinctly mysterious 'action at a distance'.

Gravitational

Electrical

Magnetic

Nuclear

Fields

To get a handle on the first three of these phenomena, we introduce the ideas of 'charge' and 'field', and we make these two general assertions:

A 'charge' generates a field in the space around it, in principle extending all the way to infinity

A 'charge' placed in a field experiences a force according to the equation Force = 'charge' x field strength

To deal with a particular instance, we attach the appropriate adjective to each of the three words in this general equation. Thus:

gravitational force = gravitational 'charge' x gravitational field strength

electric force = electric 'charge' x electric field strength

magnetic force = magnetic 'charge' x magnetic field strength

In some cases, the collections of words in these equations have special names. Here is a table summarising the situation.

Gravitational

F_G = mg

If the object in question is on the surface of the earth, then g, the gravitational field strength, has the approximate value - 9.81 Nkg^-1 and F_G is called the weight. g is negative because we measure radii of the earth as positive away from the centre, whereas g points towards the centre.

The force acts in the same direction as the field

Electrical

F_E = qE

q = electric charge, in coulombs

E = electric field, in N C^-1 or V m^-1

The force acts in the same direction as the field. If the charge is negative, then the formula will give a negative result (and a negative force in the direction of the field is, of course, actually a force in the opposite direction to the field!)

Magnetic

F_B = qvB

F_B = LiB

B = magnetic field, in Tesla

q = electric charge; v = velocity of the charge;

L = length of wire; i = current in the wire

The force acts at right angles to both the direction of the field and the direction of the current, in accordance with Fleming's Left Hand Motor Rule.

Vocabulary

Gravitational charge is usually called 'mass' (but more properly 'gravitational mass')

Electric charge is often abbreviated simply to 'charge'

In the case of magnetism, we are using a composite expression for 'magnetic charge'. One version (Li) focuses on the large scale object that can generate or experience the field, while the second version (qv) focuses on the microscopic fact that it is moving charges that constitute a current: moving charges both generate and experience magnetic fields. The two versions are interchangeable: the field can be generated by a current carrying wire, but experienced by a moving charge.

Notwithstanding the last paragraph, the various sorts of 'charge' are not in general interchangeable. So, for example, a stationary electric charge cannot feel a magnetic field, and a massive lump wood will not react to an electric field.

It is worth noting that there is fourth type of charge, called 'colour charge', that comes into play when we consider the forces between the sub-atomic particles known as quarks. In the case of mass, there is only one value (positive). In the case of electric and magnetic charges there are three values (positive, negative and neutral) while, in the case of colour charge, there are seven values (known whimsically as red, green, blue, anti-red, anti-green, anti-blue and colourless).

Now you have read this far, just look at the four equations in the table again, and note once more that they all have the pattern Force = 'charge' x field strength

So now, drilling down further, the obvious requirement is formulae for calculating the fields. Happily, once again, there is a general formula, which may be used when the field-producing charge is concentrated at a single point.

The 4πr² arises from the following consideration. The 'influence' from the 'charge' is often (but not for magnetism) depicted as a bundle of arrows spread around equally in all directions. As you go further and further from the charge, the arrows get more and more spread out, and this denotes a weakening of the field strength. In fact we can think of the field strength in terms of the number of these arrows crossing unit area at any particular point. At a distance r, the arrows are spread out over the surface of a sphere of radius r - i.e. over an area of 4πr². So the field strength is proportional to the quantity of charge divided by the area of this sphere.

Here are the equations.

Gravitational		M is the gravitational mass of the body producing the G-field. The constant G is negative and goes on top. The negative sign expresses the fact that masses always attract one another, so the G-field produced by a mass is towards itself (in the opposite direction to radius vectors radiating out from it). G is called the universal constant of gravitation, and has the value 6.7x10^-11 SI units. Note that the 4π of the general formula is incorporated as part of the constant G. This law is known as Newton's Law of Gravitation.
Electrical		Q is the electric charge producing the E-field. The constant e₀ is positive and goes on the bottom. The positive sign expresses the fact that positive charges repel, so the E-field produced by an electric charge is away from itself (in the same direction to radius vectors radiating out from it). ε₀ is called the permittivity of free space and has the value 8.85x10^-12 SI units. This formula is known as Coulomb's Law
Magnetic		I is the current flowing in a short length δL of wire that is producing the B-field. The constant μ₀ is positive and goes on top. The situation is a little more complicated in that the direction of the B-field is given by the Right Hand Grip Rule and that the variation is not radially symmetrical. μ₀ is called the permeability of free space and has the value 4πx10^-7 SI units. This formula is known as the Biot-Savart Law.

Potentials

Here is another pair of general assertions about a quantity called 'potential'.

Field strength = - potential gradient

Energy = 'charge' x potential

Once again in order to particularise these to actual equations, we have to insert the appropriate adjective before each component of the assertion. So the second one becomes

gravitational potential energy = gravitational 'charge' (i.e. mass) x gravitational potential

electric potential energy = electric 'charge' x electric potential

(We don't write an equation for the magnetic case because of the complication introduced by the direction rules for magnetic fields.)

Here is a master table containing all the relationships we have mentioned. You should verify by differentiating the potential (column 2) that the field strength (column 3) is indeed minus the potential gradient. Note carefully the minus signs when you do this - sometimes as many as three at once!

Phenomenon	Potential	Field strength	Potential energy	Force (1)	Force (2)
Gravity					F_G = mg
Electrostatics					F_E = qE
Magnetism					F_B = qvB F_B = LiB

(You should note that this discussion has only considered the situation when we have small (point) 'charges' producing the fields and equally small 'charges' experiencing the fields. There is more physics to do when the 'charges' are bigger geometrically!)