Arriving at the differential forms of Maxwell's equations

In summary: Very true! The differential forms are a more delicate way of looking at things, and are most useful for things where the integrals are not equal at some point. This is especially true for fields like the magnetic field, which can be complex and have a lot of different components.
  • #1
snoopies622
846
28
In college I learned Maxwell's equations in the integral form, and I've never been perfectly clear on where the differential forms came from. For example, using [itex] \int _{S} [/itex] and [itex] \int _{V} [/itex] as surface and volume integrals respectively and [itex] \Sigma q [/itex] as the total charge enclosed in the given volume, we can express Gauss's law as
[tex]
\int _{S} E \cdot dS = \frac {1}{\epsilon_0} \Sigma q

[/tex]
and with the divergence theorem we can replace
[tex]
\int _{S} E \cdot dS
[/tex] with [tex]
\int _{V}
\nabla \cdot E
[/tex]
and of course the total enclosed charge can be thought of as the volume integral of charge density, so
[tex]
\frac {1}{\epsilon_0} \Sigma q = \frac {1}{\epsilon_0} \ \int _{V} \rho
[/tex]

giving us
[tex]
\int _{V}
\nabla \cdot E
=
\frac {1}{\epsilon_0} \ \int _{V} \rho =
\int _{V} \frac {\rho}{\epsilon_0}

[/tex]
Since the differential form is
[tex]

\nabla \cdot E
=
\frac {\rho}{\epsilon_0}

[/tex]

does that mean it's mathematically valid to simply drop those integral symbols? I understand conceptually why that would be true, since the volume can be any non-zero size or at any location in space, it's just something I haven't seen before. Something similar happens with the other equations and the Kelvin Stokes theorem.

Thanks.
 
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  • #2
snoopies622 said:
does that mean it's mathematically valid to simply drop those integral symbols? I understand conceptually why that would be true, since the volume can be any non-zero size or at any location in space, it's just something I haven't seen before. Something similar happens with the other equations and the Kelvin Stokes theorem.

Thanks.

A rough mathematical argument is: if two functions are not equal at some point then one must be larger than the other. Assuming they change continuously, they one must be larger than the other on some finite volume, hence the integrals over this volume would not be equal.

If the integrals are equal on all volumes, therefore, the functions must be equal at every point.
 
  • #3
You can also think of the differential operators div and curl being defined as limits of integrals, which has the advantage that the definitions are manifestly covariant, i.e., independent on the coordinates choosen to evaluate them. Then your example becomes a triviality, because assuming that Gauss's Law in integral form holds for all volumes and boundaries of these volumes by just shrinking a volume to a point leads to the differential form of the equation.

As a theoretical physicist I'd also consider the differential (local) form of Maxwell's equations the fundamental laws. The integral forms are much more delicate to handle right, which is shown by the great confusion they provide due to sloppy treatment in many (otherwise maybe good) textbooks. The prime example for this is Faraday's Law, which is mostly quoted for the special case of areas and boundaries at rest and then applied without further thinking to moving areas and boundaries, leading to a lot of confusion.

The resolution of this confusion BTW finally leads to the only adequate framework for electrodynamics, which is (special) relativity.
 
  • #4
Thanks both PeroK and vanhees71, that was very helpful.

vanhees71 said:
. . which has the advantage that the definitions are manifestly covariant, i.e., independent on the coordinates choosen to evaluate them.

Ah! I was in fact also wondering about the usefulness of the differential forms. For example, with
[tex]
\int _{C} B \cdot dl = \mu _{0} i [/tex]
I can easily arrive at the magnetic field strength at a given distance r from a wire by drawing a circle around it of radius r and then dividing [itex] \mu _{0} i [/itex] by the circumference of that circle. While the differential form [itex] \nabla X B = \mu _{0} J [/itex] only tells me that the curl of the magnetic field at that distance is zero, which is the case for countless magnetic fields, including none at all.
 
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Related to Arriving at the differential forms of Maxwell's equations

What are the differential forms of Maxwell's equations?

The differential forms of Maxwell's equations are a set of four partial differential equations that describe the fundamental laws of electromagnetism. They are named after the physicist James Clerk Maxwell, who first derived them in the 1860s.

Why are the differential forms of Maxwell's equations important?

The differential forms of Maxwell's equations are important because they provide a concise and elegant way to describe the behavior of electric and magnetic fields. They are also essential for understanding and predicting the behavior of electromagnetic waves, which are crucial in modern technologies such as radio, television, and wireless communication.

How were the differential forms of Maxwell's equations derived?

The differential forms of Maxwell's equations were derived by James Clerk Maxwell using a combination of experimental observations and mathematical equations. He first combined the laws of electricity and magnetism into four equations, known as Maxwell's equations. He then used vector calculus to express these equations in their differential form.

What is the difference between the integral and differential forms of Maxwell's equations?

The integral and differential forms of Maxwell's equations are two different mathematical representations of the same fundamental laws of electromagnetism. The integral form uses surface and line integrals to describe the behavior of electric and magnetic fields, while the differential form uses partial derivatives to describe the same behavior.

What are some applications of the differential forms of Maxwell's equations?

The differential forms of Maxwell's equations have a wide range of applications in various fields, including electrical engineering, telecommunications, and physics. They are used to design and analyze electronic circuits, antennas, and other devices. They are also crucial for understanding the behavior of electromagnetic waves, which are used in technologies such as radar, GPS, and MRI machines.

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