# Solving Electrodynamics: Showing E Obeys Maxwell's Equations

• maverik
In summary, The conversation is about using Maxwell's equations to show that an electric field, given by E(r,theta,phi,t)=A(sin theta/r)(cos(kr-wt)-(1/kr)sin(kr-wt)) in the phi direction, obeys all four of Maxwell's equations in vacuum. There is a discussion about the first equation, where it is determined that the divergence of E is equal to zero. There is also a discussion about finding the associated magnetic field, B, and integrating over time. However, there is some confusion and errors in the calculations, and the need for more advice and hints is mentioned.
maverik
Electrodynamics q!

Struggling with this module, if someone could get me started in the right direction that'd be great!

An electric field is given by

E(r,theta,phi,t)=A(sin theta/r)(cos(kr-wt)-(1/kr)sin(kr-wt)) in phi direction

Show that E obeys all 4 of Maxwells equations in vacuum (no free charges and no free currents)

for Maxwells 1st eqn I said that there is only a phi component in the electric field so the divergence is d/d(phi) of E. As there is no phi term in the equation this goes to zero. Is this correct? If so how do I continue with Maxwells other equations? Any help wiould be greatly appreciated!

You can simply write down the equations using the $\nabla$ operator as you are used to. Then to do the explicit calculation, you can copy the identities from this bookmark-worthy Wikipedia page.

ok so for maxwells 1st eqn the div of E is just the phi component and gives 1/(r sin theta)d/d phi(E). Again d/d phi(E) is zero so the whole eqn goes to zero. Is this right? Also I'm not sure how to get maxwells 2nd equation as i have no term for B and when trying to calculate the 3rd eqn i get zero again as there is no phi term in E. Very confused alltogether!??

That electric field only obey all 4 Maxwell equations for a particular associated magnetic field...it's up to you to find what that field is ...You shouldn't be getting zero for $\mathbf{\nabla}\times\textbf{E}$...if you show us your calculation, we should be able to spot your error

gabbagabbahey said:
That electric field only obey all 4 Maxwell equations for a particular associated magnetic field...it's up to you to find what that field is

Hint: there is a dB/dt somewhere in the equations.

Ok now I am now getting a fairly long formula for the curl of E. I have tried letting that equal to -dB/dt (Maxwells 3rd eqn) and integrating over time to get an expression for -B. However, when i then get the div of B, nearly everything cancels but not quite everything, so i do not get zero (Maxwells 2nd eqn) (unless, this is, cos^2(theta)=2cos(theta)sin(theta), which I'm pretty sure it doesn't!). This is quite a long and tedious calculation and I am still not sure I am going down the right path. Is there any more advice or hints anyone can give!?? Again, any help would be hugely appreciated.

I suppose my main problem is finding the particular magnetic field, B, associated with the electric field given by E.

maverik said:
Ok now I am now getting a fairly long formula for the curl of E. I have tried letting that equal to -dB/dt (Maxwells 3rd eqn) and integrating over time to get an expression for -B. However, when i then get the div of B, nearly everything cancels but not quite everything, so i do not get zero (Maxwells 2nd eqn) (unless, this is, cos^2(theta)=2cos(theta)sin(theta), which I'm pretty sure it doesn't!). This is quite a long and tedious calculation and I am still not sure I am going down the right path. Is there any more advice or hints anyone can give!?? Again, any help would be hugely appreciated.

As long and tedious as it might be, the only way we 'll be able to see where you are going wrong is if you post your calculations...let's start with your result for $\mathbf{\nabla}\times\textbf{E}$...

## 1. What are Maxwell's Equations?

Maxwell's Equations are a set of four fundamental equations that describe the behavior of electric and magnetic fields. They were first proposed by James Clerk Maxwell in the 19th century and have since been widely accepted as a cornerstone of electromagnetism.

## 2. How do Maxwell's Equations relate to Electrodynamics?

Maxwell's Equations are the foundation of Electrodynamics, which is the study of the interactions between electric and magnetic fields. These equations mathematically describe the behavior of these fields, allowing us to understand and predict their behavior.

## 3. What is the significance of showing that E obeys Maxwell's Equations?

Showing that E (electric field) obeys Maxwell's Equations is significant because it confirms the validity and accuracy of these equations in describing the behavior of electric fields. It also allows us to use these equations to solve complex problems and make predictions about electric fields.

## 4. How do you solve Electrodynamics problems using Maxwell's Equations?

To solve Electrodynamics problems using Maxwell's Equations, you first need to set up the problem by identifying the relevant variables and determining which equation(s) to use. Then, you can use mathematical techniques such as integration and differentiation to solve for the desired quantities.

## 5. What are some real-world applications of solving Electrodynamics using Maxwell's Equations?

The applications of solving Electrodynamics using Maxwell's Equations are vast and diverse. They include the design and functioning of electrical circuits, the behavior of electromagnetic waves (such as radio waves and light), and the development of technologies such as wireless communication, medical imaging, and particle accelerators.

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