The Square of Electricity

The electric force F, the electric field E, the electric potential energy U and the electric potential V are all be related. The best way to see the relationship is to arrange them in a square, the square of electricity.(Bold indicates a vector)

 F U E V

Now let's look at the relations between these terms.

To find the force, F, on a charge q in an electric field, E, F = qE

To find the potential energy of a charge, q, in a potential V, U = qV

To find the potential energy change, U, from a force field, F, recall that work is force times distance and integrate U = Ú F dr, What this means is that there is a choice of reference point for the zero of potential energy.

To go the other way differentiate F = -dU/dr

To find the potential change, V, from an electric field, E, recall that work is force times distance and integrate V = Ú E dr. There is of course the need to choose a reference point for zero potential.

To go the other way differentiate E = -dV/dr

So now we can add to our square of electricity the sumary of how to go from one term to the next.

 F F =-dU/dr r/|r| U F = qE Electricity U = qV E E=-dV/dr r/|r| V

Now let's apply this to the example of two point charges.

Consider a point electric charge Q, and a second point charge q separated by a distance r.

The Force

The force between these charges is F = kQq/r^2 r/|r|

Place charge Q at the origin of coordinates then the vector r connects Q to the charge q.

r/|r| , a vector divided by its magnitude, is a way of writing a unit vector in the direction of r.

K is a constant, 9 x 10^9 Nm^2/C^2

The force between two positive charges will be radially outward from Q.

It has units of newtons.

The Field

Surrounding charge Q is an electric field filling space. The field tells the direction and magnitude of a force which would be felt by a charge q placed at a point in space.

The field is the force per unit charge.

E = F/q or F = qE

so

E = kQ/r^2 r/|r|

Think about buying hamburger, you can examine the price of the hamburger, as well as the price per pound of the hamburger. The price is like the force and the price per pound is the field.

The electric field has units of newtons per coulomb

The Potential Energy

The potential energy of our charges is U = kQq/r

The potential energy is zero when the charges are infinitely far apart. It increases and has a two charges of the same sign are brought together.

The force is the negative of the gradient of the potential energy.

F = -delU

(Your computer may not have the symbol for gradient, but here the field is purely radial so we can get away with a derivative.)

F = - dU/dr r/|r|

The electric potential energy has units of joules.

The Potential

The potential is also known as the voltage.

Just as we created a quantity which was the force per charge we are also going to create a quantity that is the potential energy per charge.

V = U/q or U = qV

V = KQ/r

The electric field is the negative of the gradient of the potential.

E= -delV

E = - dV/dr r/|r|

The potential has units of joules per coulomb also known as volts

The potential energy and the potential both involve the addition of an arbitrary constant to set the zero reference level. For point charges this constant is chosen to be zero so that the potential energy and the potential are zero at infinity.

So we can fill in our square

 F = -dU/dr r/|r| F = kQq/r^2 r/|r| U = kQq/r Electricity E = F/q k = 9x10^9 V = U/q Nm^2/C^2 E = kQ/r^2 r/|r| V = kQ/r E = -dV/dr r/|r|

 Scientific Explorations with Paul Doherty © 2008 11 January 2008