# Electromagnetic theory, lecture — III.

## Helmholtz theorem. Vector and Scalar Potentials.

“This lecture, the web version of the third lecture given in the * electromagnetic theory *paper of the physics honors degree class, was delivered on 9th January 2018.

**All electromagnetic theory lectures of this series, will be found here. “**

**Topics covered**

**A.** **Formalism of electrodynamics** — fundamental theorem

**B.** **Application of Helmholtz theorem** — to electrostatics

**C.** **Application of Helmholtz theorem** — to magnetostatics

## Formalism of electrodynamics

### Helmholtz theorem

In our previous lectures, lecture – I and lecture — II, we discussed the * Maxwell’s equations *in

**free space**and in

*. We also discussed the boundary conditions which helps us solve the and fields.*

**material media**The and fields are vector fields and Maxwell equations involve *differentiation* of these vectors. So we have to deal with * curl* — and

*— , of these vectors — or vector fields, apart from any*

**divergence***to be satisfied by these, as we saw in our last lecture linked.*

**boundary conditions**The * Helmholtz theorems* provides some useful

*relations*and

*properties*which helps us determine the and fields

*uniquely*. It is important to understand the mathematical rigor and the physical vigor these equations provide in having an unified formal treatment that all of

*electrodynamics*espouses.

One needs a basic proficiency in vector calculus to be able to grasp the inherent significance of these methods. For the beginning you can go through the slides here — * vector calculus*, to have a reasonable hold on the vector calculus concepts.

*There is a theorem which deals with the implications of curl of a vector.*

### Theorem 1

If curl of a * vector field* is

*zero*— everywhere, then can be written as gradient — , of a

*, also known as*

**scalar field***scalar*

*.*

**potential function***Here minus sign is just a convention. *

**a.** Such fields — here, are known as * curl-less* or

*.*

**“irrotational” fields****b.** The curl is *zero* *everywhere*. I.e.

**c. **The quantity is independent of path — of *integration*, as long as and are fixed.

**d.** The quantity for * closed paths* or loops of integration.

**e. **The vector can be written as a * gradient of some scalar* such that, .

**f.** If any one of the above statement is valid, it ensures all the others to be valid. That implies and so on.

**g.** Scalar potential is * not* unique. Any function independent of position can be added to without changing

*Now there is a theorem which deals with the implications of divergence of a vector.*

### Theorem 2

If divergence — , of a * vector field* is

*zero*— everywhere, then can be expressed as the curl of a

*.*

**vector potential function****a.** Such fields — here, are known as * divergence-less* or

*.*

**“solenoidal” fields****b.** The divergence is *zero* *everywhere*. I.e. .

**c. **The quantity — is independent of *surface* of *integration*, as long as the boundary of the surface — a line, is fixed.

**d.** The quantity for * closed surface* of integration.

**e. **The vector can be written as the * curl of some vector* such that, .

**f.** If * any* one of the above statement is valid, it ensures all the others to be valid. That implies and so on.

**g.** Vector potential is * not* unique as

*gradients of any scalar function is curl-less*. Gradients of any scalar function can be added to the vector and this transformation would leave the vector unaltered. I.e. if both new potential and old potential would give rise to the same .

There is another mathematical result which has implications for both of the above theorems and we will state it.

Any general vector irrespective of whether this vector is divergence-less or *not*, and whether this vector is curl-less or *not*, can always be written as * sum of two vectors*,

**a.** one vector which is the * curl of some vector* and

**b.** another vector which is the * divergence of some scalar function*.

In symbols: .

## Application of the above theorems to 3 situations

We will study how the above theorems have a great deal of implication for almost all of the field of electromagnetic theory.

But we will study them under 3 situations — gradually moving up the ladder of generality, beginning with our familiar knowledge of the static fields.

**a. Electrostatics **

**b. Magnetostatics **

and

**c. Electrodynamics**

### Electrostatics

When electric charges are stationary the resulting electric fields lead to electrostatic behavior. Under these conditions electric fields — , are always “irrotational“, i.e. *curl-less*, thus they satisfy conditions of — * theorem 1, of Helmholtz theorem*.

Because , all the other conditions are satisfied and a potential function known as * scalar potential* () can be defined.

Here Ref — for reference, on the lower limit of the integral sign stands for a particularly chosen location for the potential, from where all calculations are to be made. There are *two standards* for this idea to be implemented;

**a.** if * charge distributions are finite*, reference can be set to

**b.** if * charge distributions are not finite*, other means are to be explored to calculate E.g. one should

*set on the lower limit of integration, in place of Ref, for an*

**not***infinitely long cylinder*.

The above boxed equation which we can define because *curl of the electrostatic field* is *zero*, is an * integral form* of the scalar potential, . The

*of definition of*

**differential form***is then; .*

**scalar potential**In the electrostatics conditions, potential formulation serves a very *useful* role, as 1 scalar equation is to be solved in lieu of 3 vector equations — we do not have to bother about the 3 vector components which makes the problem cumbersome.

The two Maxwell equations — one vector and one scalar, and which are 4 equations in total give rise to Poisson’s equation.

* Poisson’s equation*:

When source term is absent i.e. , Poisson’s equation turns into Laplace equation.

* Laplace equation*:

Potential can now be calculated as:

where , is known as the * separation vector*. Now electric field can be calculated from The expression is nothing but the

*.*

**Coulomb’s law**### Magnetostatics

* Magnetostatics *problems follow a similar logic. The magnetostatic nature is defined by

*or*

**stationary currents***. That means charge densities that produce the field must not vary in time.*

**steady-currents**Mathematically: . When used in the equation of continuity: , this means: . Thus the * condition for magnetostatics* is given by: .

*The second Maxwell equation is: , i.e. magnetic fields are divergence-less in all situations. *

According to * theorem 2 of Helmholtz theorem* then, magnetic field can always be written as curl of a vector potential , i.e. .

is known as vector potential or * magnetic vector potential*.

By * Ampere’s law of Maxwell equations* — i.e. , we have: .

We already discussed that the magnetic vector potential is not unique. * Gradient of any scalar is curl-less*.

By this * freedom *we can always scalar , so that . If original vector potential is not divergence-less we add to it, to make it divergence-less.

Thus, and .

For we have . This is nothing but Poisson’s equation, as we have seen.

Remember Poisson’s equation for electrostatics: . So behaves like source: .

As in the case of electrostatics case if *we have a solution to the Poisson’s equation*: .

This * solution *is given by: .

If the source term does not go to *zero*, (at ) we have to use other means as in the case of electrostatics.

It is always possible to find , to make divergence-less.

Thus Ampere’s law becomes: . This is also Poisson’s equation — actually 3 Poisson’s equations, because its a vector equation.

If the solution is:

Thus for steady currents we have

We will discuss the 3rd situation of applicability of the Helmholtz theorems to the most general case of electrodynamics in the next lecture.

Categories: basic physics, Courses I developed, Electromagnetic theory, Physics, Teaching

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