# Understanding Maxwell's Equations

• robert25pl
In summary, free space means that: \rho=0 and \vec\jmath=0. So, in two of Maxwell's Equations, this means that:\nabla \cdot \vec E=\rho ={\color{red}\ 0} and\nabla \times \vec B=\vec \jmath + \frac{\partial \vec E}{\partial t}={\color{red}\ \vec 0} + \frac{\partial \vec E}{\partial t} .
robert25pl
I want to make sure that I understand this good.
Given E and B are possible in a region of free space (J=0) only if $$\triangledown \times E=0$$ and $$\triangledown \cdot B = 0$$

That $\vec{B}$ needs to be stationary (time independent)...Else $\vec{E}$ would not be a purely potential-derived field.

Daniel.

I have this two equations:

$$E=3\sin(3z-6t) \vec{k}$$
$$B=- \frac{1}{15} \sin(3z-6t) \vec{j}$$

So what should I do first?

Verify whether such a field configuration satisfies the eqn-s

$$\nabla\cdot\vec{B}=0$$

$$\nabla\times\vec{E}=-\frac{\partial \vec{B}}{\partial t}$$

Isn't this what u're supposed to do?

Daniel.

robert25pl said:
So what should I do first?

That depends on the question you're supposed to be answering.

"Free space" means $$\rho={\color{red}0}$$ and $$\vec\jmath={\color{red}\vec 0}$$.
So, in two of Maxwell's Equations, this means that
$$\nabla \cdot \vec E=\rho ={\color{red}\ 0}$$ and
$$\nabla \times \vec B=\vec \jmath + \frac{\partial \vec E}{\partial t}={\color{red}\ \vec 0} + \frac{\partial \vec E}{\partial t}$$.
Of course, [from the other two equations] we must always have
$$\nabla \cdot \vec B= 0$$ and
$$\nabla \times \vec E= -\frac{\partial \vec B}{\partial t}$$

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What?! When we were taught Maxwell's equations in free space, we were told that: $$\nabla \cdot \vec E=\frac{\rho}{\epsilon_{0}}$$, as free space meant in air not in any sort of medium.

Nylex said:
What?! When we were taught Maxwell's equations in free space, we were told that: $$\nabla \cdot \vec E=\frac{\rho}{\epsilon_{0}}$$, as free space meant in air not in any sort of medium.

So, maybe term I should have used is "source-free".

$$\nabla\cdot\vec{B}=0$$ I verified that

$$\nabla\times\vec{E}=\left|\begin{array}{ccc}\vec{i }& \vec{j} &\vec{k}\\\frac{\partial}{\partial x}& \frac{\partial}{\partial y}& \frac{\partial}{\partial z}\\ 0 & 0 & 3\sin (3z-6t) \end{array} \right|$$

$$\triangledown \times E$$ gave me 0 and $$-\frac{\partial \vec{B}}{\partial t} =- \frac{2}{5} \cos(3z-6t) \vec{j}$$

I think this wrong?

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robert,

You say:

divB = 0, I agree

curlE = 0, I agree

-dB/dt = (-2/5)sin(3z - 6t)j, I think you've got a mistake here

Should be a "cos" there.

Daniel.

$$\frac{\partial \vec{B}}{\partial t} =-\frac{2}{5} \cos(3z-6t) \vec{j}$$
So $$\nabla \times \vec E= -\frac{\partial \vec B}{\partial t}$$
are nor equal and they are not possible in region of space?

That's that.It is not possible.They should identically solve every equation from Maxwell's system...

Daniel.

## 1. What are Maxwell's Equations and why are they important in science?

Maxwell's Equations are a set of four mathematical equations that describe the fundamental principles of electricity and magnetism. They are important because they provide a complete and accurate description of how electric and magnetic fields behave and interact with each other, which is crucial for understanding many natural phenomena and developing technologies.

## 2. How were Maxwell's Equations developed?

Maxwell's Equations were developed by Scottish physicist James Clerk Maxwell in the 1860s. He combined and expanded upon the work of previous scientists, such as Michael Faraday and André-Marie Ampère, to formulate a unified theory of electromagnetism.

## 3. Can you explain each of the four equations in simpler terms?

The first equation, Gauss's Law, states that the electric flux through a closed surface is proportional to the charge enclosed by that surface. The second equation, Gauss's Law for Magnetism, states that there are no magnetic monopoles and that magnetic field lines always form closed loops. The third equation, Faraday's Law, states that a changing magnetic field induces an electric field. And the fourth equation, Ampère's Law, states that a changing electric field induces a magnetic field.

## 4. What are some practical applications of Maxwell's Equations?

Maxwell's Equations have numerous practical applications in modern technology, including the development of wireless communication, electric motors, generators, and transformers. They are also used in the design of antennas, radar systems, and optical devices.

## 5. Are Maxwell's Equations still valid today?

Yes, Maxwell's Equations are still considered to be valid and accurate descriptions of electromagnetism. They have been extensively tested and verified through experiments and are used as the foundation for many theories and technologies in physics and engineering.

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