# Maxwell Equations

### Basic Concepts and Principles

Maxwell's equations are partial differential equations governing electromagnetic phenomena in vacuum.
These equations are

**General Maxwell's Equations**

1) \( div\vec{E} = \frac{1}{\epsilon_{0}\rho} \)

2) \( div\vec{B} = 0 \)

3) \( rot\vec{B} = \frac{\partial \vec{B}}{\partial t} \)

4)\( rot\vec{B} = \mu_{0}\vec{j}+\epsilon_{0}\mu_{0}\frac{\partial \vec{E}}{\partial t} \)

**In absence of electric charges and currents**

1) \( div\vec{E} = 0 \)

2) \( div\vec{B} = 0 \)

3) \( rot\vec{B} = \frac{\partial \vec{B}}{\partial t} \)

4) \( rot\vec{B} = \epsilon_{0}\mu_{0}\frac{\partial \vec{E}}{\partial t} \)

Where E y B are the intensity of electric and magnetic fields respectively, j is the electric current density, ρ is the charge density, and ε0,
μ0 are the constants

$$ \epsilon_{0}=8,854.10^{-11} m^{-2}.N^{-1}.C^{2}$$ $$ \mu_{0}=1,257.10^{-6} mk.C^{-2}$$

These two constants are related to the speed of light in vacuum, c, exactly

$$ c^{2}=\frac{1}{\epsilon_{0}\mu_{0}}$$

$$ \epsilon_{0}=8,854.10^{-11} m^{-2}.N^{-1}.C^{2}$$ $$ \mu_{0}=1,257.10^{-6} mk.C^{-2}$$

These two constants are related to the speed of light in vacuum, c, exactly

$$ c^{2}=\frac{1}{\epsilon_{0}\mu_{0}}$$

If we take Maxwell's equations in a vacuum, we can obtain the wave's equation as follows: first, we derive for t in last one and substituting in the second, this provides

$$ \epsilon_{0} \mu_{0} \frac{\partial^{2} \vec{E}}{\partial t^{2}} = rot \: rot \vec{E} $$

Using the relation

$$\Delta \vec{F} = \triangledown (div\vec{F}) - rot \: rot \vec{F} \Rightarrow rot \: rot \vec{F} = \triangledown (div\vec{F}) -\Delta \vec{F} $$

So we obtain

$$ \epsilon_{0} \mu_{0} \frac{\partial^{2} \vec{E}}{\partial t^{2}} = \triangledown (div\vec{E}) -\Delta \vec{E} $$

Applying the first of equations (in vacuum) we finally obtain the wave's equation

$$ \epsilon_{0} \mu_{0} \frac{\partial^{2} \vec{E}}{\partial t^{2}} = rot \: rot \vec{E} $$

Using the relation

$$\Delta \vec{F} = \triangledown (div\vec{F}) - rot \: rot \vec{F} \Rightarrow rot \: rot \vec{F} = \triangledown (div\vec{F}) -\Delta \vec{F} $$

So we obtain

$$ \epsilon_{0} \mu_{0} \frac{\partial^{2} \vec{E}}{\partial t^{2}} = \triangledown (div\vec{E}) -\Delta \vec{E} $$

Applying the first of equations (in vacuum) we finally obtain the wave's equation

$$ \epsilon_{0} \mu_{0} \frac{\partial^{2} \vec{E}}{\partial t^{2}} = c^{2} \Delta \vec{E} $$

From the above, if for an observer the speed of light is not c then Maxwell's equations could not be hold for him.

Suppose now two inertial observers, called A and B. Suppose A moving uniformely to half speed of light in a radial direction to B and assume that B observer turns on a flashlight directed to A.

B notes that the the ray of light away from him at a speed c = 300.000 km/s.

Logically, A should measure the ray of light travels at a speed

$$c - \frac{c}{2} = \frac{c}{2} = 150.000 km/s (aprox.) $$

This would mean that for observer A, Maxwell's equations fails, at least as way we have seen before, because ligth-speed is not c. However, A observer might say that he is no moving at all. Who is moving is B in opposite direction!!.

Moreover, if A were moving to B he could measure the speed of light even higher than c!!!

In an attempt to explain this discrepancy, scientists of the nineteenth century created the "ether light" theory . On This theory speed of light in vacuum would only make sense respect to ether, this, in an analogy with the sound on earth whose speed of 340m / s is only meaningful relative to air.

Several experiments (like Michelson-Morley experiment in 1887) showed that this substance could not exist. Thus, classical mechanics and in particular the kinematics should be reviewed.

The special relativity theory was born to solve these (and other) questions, teaching us that there is no (inertial) observer privileged and for they all, speed of light is invariantly equals to 300.000km / s.

We will see the formulas in detail, but anticipating we will say now that time becomes relative to observer who measure it. As we moving approaching speed of light, our clocks goes moving slower, ie the time dilates and space coordinates suffer a expansion.

Suppose now two inertial observers, called A and B. Suppose A moving uniformely to half speed of light in a radial direction to B and assume that B observer turns on a flashlight directed to A.

B notes that the the ray of light away from him at a speed c = 300.000 km/s.

Logically, A should measure the ray of light travels at a speed

$$c - \frac{c}{2} = \frac{c}{2} = 150.000 km/s (aprox.) $$

This would mean that for observer A, Maxwell's equations fails, at least as way we have seen before, because ligth-speed is not c. However, A observer might say that he is no moving at all. Who is moving is B in opposite direction!!.

Moreover, if A were moving to B he could measure the speed of light even higher than c!!!

In an attempt to explain this discrepancy, scientists of the nineteenth century created the "ether light" theory . On This theory speed of light in vacuum would only make sense respect to ether, this, in an analogy with the sound on earth whose speed of 340m / s is only meaningful relative to air.

Several experiments (like Michelson-Morley experiment in 1887) showed that this substance could not exist. Thus, classical mechanics and in particular the kinematics should be reviewed.

The special relativity theory was born to solve these (and other) questions, teaching us that there is no (inertial) observer privileged and for they all, speed of light is invariantly equals to 300.000km / s.

We will see the formulas in detail, but anticipating we will say now that time becomes relative to observer who measure it. As we moving approaching speed of light, our clocks goes moving slower, ie the time dilates and space coordinates suffer a expansion.

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### Post from other users

### Mathstools Administrator:

2012-12-02 11:51:12Absolutelly, Berndt. You are right, we change this. Thanks

### Berndt Barkholz:

2012-12-01 03:53:18You write (Maxwells equations) : c = 1/ (ε0 μ0) I hope you know this is wrong, it should be : c = 1/ (ε0 μ0)^0.5 or c^2 = 1/ (ε0 μ0)..... best regards

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