Grad, div and curl explanations

[tommyers]

Hi,

I am teaching myself electrostatics – something which seems to have been ‘skipped’ over in my electronics degree. I am working from the book: Fudamentals of engineering electromagnetics’ by David K. Cheng.

My particular interest is electrostatics – more specifically the study of capacitance and capacitor geometry.

My studies have brought me to the gradient, divergence and curl. Below are my queries / observations for each:

Gradient
The term ‘space-rate’ of a scalar field seems to appear often. Does this just relate to the x,y,z and time of the scalar field? i.e the x,y,z relate to the space and the time part to the rate?

A diagram on P39 of the book shows two parallel surfaces. The lower surface is labelled, V and the upper V + dV, where dv indicates a small change in V. A point p1 is shown on the lower surface and point p2 is shown on the upper, normal, dn, to p1. p3 is shown close to p2 along a line, dl, however dl not equal to dn.

It states the ‘space-rate’ change dv/dl is greatest when along dn i.e the shortest distance between the two surfaces. Since the magnitude of dv/dl depends on the direction of dl, therefore it is a directional derivative.

It then states: “we define the vector that represents both magnitude and direction of the maximum space rate of increase of a scalar as the gradient of that scalar.”

They then give: grad V = an (dv/dl)

But what does this really tell me? That the gradient between points indicates the field strength between the two? In actually calculations would you have to include charge quantities?

Does anyone have any ‘real life’ examples with in depth maths to show how I can use this in my understanding and utilisation of electrostatics? Any good websites? Etc…

Divergence
The section on divergence discusses flux lines and there densities in relation to the field strength and direction….
… if an enclosed surface contains a source or sink then there will be an inward or outward flux flow. A positive net divergence indicates a source inside the volume and a net negative flow indicates a sink.

This then leads onto solutions using the ‘taylor series’.

What does the divergence tell us? Again any worked examples etc.. further description / discussion.

I have only just started on ‘curl’ but already it has begun talking about a ‘vortex source’!

I thought that some discursive explanation of the mathematics and concepts behind these 3 theorems may give me some better understanding.

Regards

Tom

 

[Integral]

Where do we start? What level of math have you been exposed to? Have you completed a calculus course?

Grad, Curl and Div are key partial differential operators, if you have not seen them before you need to find a Multivariable calculus text, this is where they are generally introduced.

 

[Doc Al]

I recommend that you check out the book "Div, Grad, Curl, and All That: An Informal Text on Vector Calculus" by H. M. Schey. It might be just what you need.

But as Integral says, it all depends on your level of math.

 

[BoTemp]

Quick explanations and examples:

Picture a hill, with some height z= f(x,y). grad z will be a vector which always points normal to the surface of the hill. As in, picture an infinitesmal square around (xo,yo). grad z at (xo,yo) will point straight away from the square. That's not it's only significance, what you mentioned above is true, it's just harder to visualize.

Picture a cylindrical pipe with water flowing through it. The flux of particles through a cross section will be velocity*time interval*area of cross section*particle density of water. Now picture a closed surface, as in a cylinder with the same shape as the pipe but a smaller length. The flux through that volume will be the sum of fluxes through each surface; in this case the total will be 0, since opposite faces cancel and the sides have no flux (assuming water isn't leaking out the pipe).

Divergence is flux/volume. For a lightbulb, one can picture the flux of photons through the glass, divide that by the volume of the bulb. Of course that's not the most general solution, must as density isn't necessarily the same throughout a material.

Curl is similar; I've seen it described as vortex density. Think of water in a vortex. It would take you work to swim around in a circle to get back to where you started. That's a path integral. The curl is a density measure of that.

This isn't the most rigorous stuff in the world, but conceptually it's correct.

 

[jtbell]

Picture a hill, with some height z= f(x,y). grad z will be a vector which always points normal to the surface of the hill.

No, grad z is the horizontal direction along which the surface of the hill has its steepest upward slope. If you put some water on the hill at point (x,y), it runs downhill in the direction given by the negative of grad z.

I suspect that you're thinking of the contour lines with constant elevation on a topographical map. On the map, the vector arrows representing the gradient are normal (perpendicular) to the contour lines at every point.

 

[neurocomp2003]

mathworld.com

 

[Daverz]

The original poster may have moved on by now, but I think the Feynman Lectures vol. 2 does a nice job explaining div, grad, and curl.

 

[tommyers]

Thank you all for your responses.

I have taken Doc Al recommendation and bought "Div, Grad, Curl, and All That: An Informal Text on Vector Calculus" by H. M. Schey.

From this and my other books I understand the following:

Gauss' Law is a general method for determining the electric field generated by a charge. The law does not give you the absolute value of E, but the flux of E, the amount of electric field 'flowing' through some surface.

This lead me onto Divergence (which I am still a little unsure) but it seems to me that it is the ratio of the surface integral to the volume enclosed by the surface, S.

Does this mean then if you have a charge, q and place some shape around it the if you integrate that shapes surface and compare it with its volume it will somehow tell you how much the charge has diverged - logic tells me that the greater distance you are from the charge, q then the greater the divergence.

I am an electronics engineering student and find vectors hard to imagine in real life situations. What does the divergence offer me practically?

My goal is this:

I have two parallel plates (forming a capacitor) I know the separation between them, d and I know the plate area, A. I would like to predict the capacitance these two form, I have used the elementary equation in the past (C = (epsilon*A)/d) but this does not give me a accurate enough answer when comparing this to my experimental results, as it neglects fringing fields! I have modeled the capacitor using a boundary element method - but would have liked to also proved my results using a purer physics method i.e gauss / maxwell and the div and grad that goes with that. If someone you provide a worked solution for me for this arrangement then that might help some of the theory stick!

Regards

Tom

 

[lalbatros]

Tom,

The (intergral) Gauss law tell us that the flux of the electric field through a closed surface is equal to total charge inside the surface (divided by e0). Because this is valid for any closed surface, it can be proved that this completely determines the electric field.

Now, if you apply the Gauss law to an infinitesimal cube, you will see that the consequence is the (local) Gauss law that say div E = rho/e0 . That's why the div operator enters the electrostatics. The other operators comes with the rest of the electromagnetic theory in a similar way.

This shows that -theoretically- electromagnetism is based on these mathematics. And -again theoreticall- electronics is based on still more than that since at quantum mechanics (semiconductors) and classical mechanics (electron tubes) may be needed.

Learning all that or not depends on the objectives. If you want to know everything about electronics, then you certainly want to know about the maths too. Some applications really need these maths, I think, for example: wave guides, radars, and indeed electrostatics.

In particular if you really need to calculate details of the fringind fields you need this kind of background. Note however that some way to solve some problems may skip these maths. For example the BEM in electrostatics does not need more math than what the field of an electric charge, the sum of potential and maybe vectors, integration, and some numerics.

You need to think about it and ask more questions if needed. However, consider that learning all that is a matter of years and not weeks.

Michel

 

[tommyers]

Hi Michel,

Thank you for your continued support.

consider that learning all that is a matter of years and not weeks.

I think this is why the project feels its becoming daunting!

And that's why I keep explaining what I want to you guys - because I feel that you know loads of the background physics behind a project like this, and you could tell me the parts I need to know, instead of me at this moment trying to learn everything, only to dig out the small areas that I need. I really need to have my models complete in the next 3 weeks so I can continue with the electronic circuitary and software that goes with this project!

How important is it that I consider Gauss' law, div and grad in this project - I was even going to try and have a brief look at conformal mapping as well as my BEM model. I just need something that will allow me to calculate the field lines around my charged plates - I my BEM model I know the charges on the plates!

I will email you with some of my results as this my help in validating my model.

Regards

Tom

 

[lalbatros]

Hi Tom,

How important is it that I consider Gauss' law, div and grad in this project

This all depends on the specifications of your project, we can only guess.
My guess is that it is not important.

Also, as I told you earlier (other threads), I even believe that this "fringing field" topic does not directly matter. I assume that your main goal is to measure the relative permittivity of some materials. Therefore you "only" need to measure a reference in air and compare to the measurement with the material under test.

The fringing fields however could introduce some imprecision in you measurement. This is why you should know about fringing and how it work physically. You mentioned sometimes that you indented to measure some granular materials. It is clear that it that case fringing fields would introduce errors. The size of the granules should at least be much smaller than the volume where the fringing fields conscentrate (near the edges). This is not easy to evaluate. But is would be easier to reduce the fringing fields and avoid this problem: simply avoid sharp edges, make then round with a large radius.

Concerning the minimum theory required. I think you need to know the following:

the q / (4 pi e0 r) law for the potential of an electric charge
the q / (4 pi e0 r²) law for the electric field, and its direction
what a perfect conductor is and why there a conductor is equipotential
why a conductor cannot have an elecric field parallel to its surface, only perpendicular
why charges accumulate in edges of plates
why the charge accumulation is reduce when two plates of a capacitor are very close to each other
what the permittivity of a material is
why a material can have a permitivity > 1
plus all theory related to measuring small capacities correctly​

On the experimental side, you should be able to proof that your device measures what you indend to measure without excessive perturbation.

Now, concerning the fringing fields effect on capacities. This is not essential (see above), but if you are interrest (this allowed of course!) you could have an efficient solution by reading the figure 12 in the paper by Erikson (http://www.ttc-cmc.net/~fme/plates.12-24-00.ps.gz" ). You could even try to make a curve of the ratio (real capacity)/(ideal capcacity) from this figure. You could check with your measurements.

Then, try to avoid edge effect instead of calculating them!

Regards,

Michel