//This?chapter?covers a number of mathematical topics that will be critical in our treatment of electromagnetism.

//Let's start it.

//And this time I won't follow the order of our textbook, for this will be too slow.

//Because I write this assuming that all readers have already known some basic concepts?of field theory, such as gradiant, divergence, curl of a scalar of vector field.

1. Definition

We know?according to Columb's law, we can prove that in the electric field of a single point charge, the work done to move another charge is only determined by its initial and ultimate position, which means for different paths, we have

and so we can prove the line integral of? around any closed path in an electrostatic field is

But this is only true for electrostatic field. We will learn more about circuit integral of electric field?in?the following chapters. So?we know

and that means? is the gradient of another scalar field:

where? is called electric potential.

And when we move a charge? from A to B, we can?find the?change of its electric potential energy is

For a single point charge?，we often put zero potential at infinite distance, and

2. Superposition principle

The potential function is simply the sum of the potential functions that we would have for each of the sources present alone- providing we make a consistent assignment of the zero of the potential in each case. And that means for a charge distribution, we have

3. Poisson's equation

We've already known:

So...

This is Poisson's equation,?relates the charge density to the potential.?We will learn?advanced versions of this equation many?chapters later, considering electromangetic induction and theory of relativity.?

If in free space where charge density is 0, we have

And this is Laplace's equation. And with this equation we can prove that: In free space there can't exist any electric potential extremum. So a charge can't find a stable equilibrium in electrostatic field.

4?Dipole

A dipole is two charges with equal and opposite charge?,?placed at? as shown in figure. In this system, using polar coordinates,?electric potential:

If we observe it from very far, which means?, we can use approximation

Actually, if we want to create a charged object that can exactly?generate an electric potential field as above, it can be a spherical shell with surface charge density distribution

And this is equal to two spheres with opposite uniform charge density placed very close. This will be useful.

If we want to calculate electric field generated by a dipole, we can simply put

And if we define dipole moment:

Field around a dipole can be expressed as

And this dipole moment is not only defined to simplify the expressions above. It can be used to find force and torque acted on a dipole:

And we will see more about?dipoles in Magnetostatics.

References

[1] Edward M. Purcell, David J. Morin. Electricity and Magnetism (Third Edition)[M]. Cambridge Univercity Press.