DI/dt depending on v?

[kolahal_b]

As we know,a single moving charge creats a current J=rho*v=qv^2*delta(r-vt) supposing that the charge moves along x axis.However,it is not a steady current.Making I=J.dS,how can we calculate the dI/dt?I suppose,it will depend on v in such a manner that if v increases,dI/dt will increase...Any help?

[ronthepon]

Why won't the current be steady? Agreed, the electric fiux linked with any stationary loop will change, but the current won't change at all.

[kolahal_b]

WHatever,my question is if dI/dt depends on v?

[ronthepon]

Oho, if v refres to 'velocity' then most definitely.

[kolahal_b]

Can you formulate it mathematically?I see that once v is high,dt in dI/dt is small and dI/dt is high enough.

[ronthepon]

I had read about the displacement current. From what I know, I'ts related to describing the magnetic feild, but it could hold importance here. I'll go to it after a little time.

 

For now, if we assume that the reference loop, through which the charge passes, is having a width of 'w'. Alternatively, we can say that the charged object has a x axis length of 'w'.

 

So, if it's travelling with a velocity 'v', it passes through the loop in time= [math]\frac{w}{v}[/math]. That'll be twice dt (as it rises to a maximum, then reduces?)

 

This concept is really really bad. I don't think it is useful. But the displacement current has an expression:

 

[math]I_{disp} = \epsilon_o \frac{d \phi_e}{dt}[/math]

I'll assume that you're relatively well versed with the symbol jargon.

 

You'll have to assume the dimentions of the reference loop, and place it on say... with the center on origin. Saying it's a circle, the electric flux linked with it can be found out by the use of integral calculus.

 

You can then easily differentiate the term you get, simply using [math]\frac{dx}{dt}=v[/math] Pretty straingforwad, but could take some hard work.

 

Once you get the expression for displacement current, simply differentiate it again.

 

Tell me if you do it... I'm recovering from a severe attack of 'exam fatigue syndrome', and need a good basketball match before I get back to the equations.

[kolahal_b]

I really dont understand why to refer to loops/displacement currents and all that!Take a charge moving and say whether dI/dt depends on v,its vel.

[ronthepon]

Unfortunately that's not possible. Firstly, how do you define current? If you define it as the charge flowing per unit time, then I'll ask "flowing through what?"

 

So you'll talk about an imaginary circle in space... that's about the loop.

 

Secondly, you'll have to give the loop as well as the charge some linear dimentions, otherwise dI/dt will be infinity as the charge instantly enters and leaves the 'loop'.

 

If you don't want the concept of displacement current, well... then I guess this the best I can do.

 

The problem with physics is that you can't ask a question using textbook jargon to anybody but a classmate. Where does this problem crop up? What do you want this concept for?

 

How exactly can I help?

[kolahal_b]

Well,I have posted in the very first thread that a charge moving along x axis constitutes a current given by J=rho*v=q*v*delta(r-vt)i where delta stands for a delta function.If you find any problem,then visualise this as a charge flowing through an infinite straight wire along x axis.That will do as well.I have talked to David J. Griffiths and confirmed that it is the case,so do not waste more tiome in talking over this matter.

Also,it's not correct that in physics,you have to have a loop/circuit to visualise currents.To derive curl of B® or divB we use a continuous current distribution.In that case,you will fail to answer what is the current actually passing through.

Now,for your interest,I am writing the real case of interest:however,do not answer if you do not understand what I say.

I was reading Griffiths where he says-to be able to get sufficiently close results in electrodynamics,using Ampere's law/Bio-Savart's law(tools of Magnetostatics) one must be operating in Quasi-static region.This means if the appreciable change in current I occurs in time t,then we must be interested in regions where distance s from the current distribution is s<<ct.In this range of s the interaction is appreciably instantaneous and we need not consider the speed of propagation of EM information.

This means that if t is very small,(i.e. the current is varying too rapidly),the relation should be treated with care.specifically,I was interested what would happen if a single point charge constitues the current.As we know a moving charge cannot create a steady current(Griffiths tells so,and by my question,I was trying to mathematically formulate this).It appears from intuition that if the speed is too high,dI/dt,evaluated at a point is high.(Because dt is very small in that case).So,in that case,t is small,if my intuition is correct.Then what would happen if the speed of the moving charge is high,say,(1/3)rd c?Are we sure in that case normal electrodynamics(Bio-Savart's law,Ampere's law etc can be operated as usual?Will this charge will serve as a "gentleman"test charge in Coulomb's law in electrostatics?I feel there is some subtlity herein as I am suspecting.

[LaurieAG]

..visualise this as a charge flowing through an infinite straight wire along x axis..... I was interested what would happen if a single point charge constitues the current..... by my question,I was trying to mathematically formulate this...

 

Hello kolahal,

 

Wouldn't the differential be the area under the curve if you had a fixed measurement point and the charge flowed over time (along the x axis)?

[kolahal_b]

What differential are you referring to?I do not understand what you say.OK,you answer to this question.Can we define the electrostatic potwential if the test charge moves at velocity 1/3rd or 1/5th of c?

[ronthepon]

Okay then. I'll be more direct.

 

Yes, for a point charge, you need a loop through which it passes for current to be a meaningful phenomenon. What you're thinking about is probably a space charge distribution. Only that gives rise to a quantity called current density.

 

Doesn't J correspond to current density? That is the current flowing through a loop, divided by the effective area of the loop perpendicular to the current.

 

Agreed, to use Biot-Savart's law, we consider a continuous charge flow, but that's only becuase we assume that there are so many charge carriers that they needn't be distinguished from each other.

 

And to use the conventional concepts does require that the charges be moving somewhat slow, as compared to the speed of light.

 

And, I might be wrong, but I don't think that you really mean Dirac's delta function. Is it the generally used term for 'change in'?

 

And in J=rho*v, what is rho?

[kolahal_b]

rho is the volume charge density.

Yes,I meant Dirac's delta Function.

My question is what would happen if the velocity of test charge is high?In other words why the charge moving with high velocity is non-conventional?

I have defined J as current density and I=integral J.dS

Continuing the discussion regarding the justification of the current by a single point charge moving in free space is not going to help.So,leave it.

[Qfwfq]

As we know,a single moving charge creats a current J=rho*v=qv^2*delta(r-vt) supposing that the charge moves along x axis.However,it is not a steady current.Making I=J.dS,how can we calculate the dI/dt?I suppose,it will depend on v in such a manner that if v increases,dI/dt will increase...Any help?

I meant to get back to this thread Friday but I coudn't. I think Kolahal that you're needing to use the derivative of Dirac's delta (as well as the chain rule, of course). I needed to shake a few cobwebs out of my memories, so I wasn't 100% sure, but the derivative of the delta is:

 

[math]\frac{d}{dx}\delta(x)=-\frac{\delta(x)}{x}[/math]

 

and the page Ron links to confirms this. However I'm not sure why you have that v squared. As far as I can remember and figure, a point charge has density [math]\norm q\delta(x)[/math] and wouldn't be proportional to v at all, so I would write:

 

[math]J=\rho v=qv\delta(r-vt)[/math]

 

from which I would get:

 

[math]\frac{dJ}{dt}=\frac{qv^2}{(r-vt)}\delta(r-vt)[/math]

[kolahal_b]

Qfwfq,I thank you very much.Well,yes!I wrongly wrote v^2 in the expression of J.However,I knew the identity you have used.But,could not work out.

Well,below I reproduce something interesting:

I was reading Griffiths where he says-to be able to get sufficiently close results in electrodynamics,using Ampere's law/Bio-Savart's law(tools of Magnetostatics) one must be operating in Quasi-static region.This means if the appreciable change in current I occurs in time "t",then we must be interested speed of propagation of EM information.

This means that if t is vin regions where distance s from the current distribution is s<<c"t".In this range of s the interaction is appreciably instantaneous and we need not consider the ery small,(i.e. the current is varying too rapidly),the relation should be treated with care.

Specifically,I was interested what would happen if a single point charge constitues the current.As we know a moving charge cannot create a steady current and you have found dJ/dt.It's proportional to v^2.So,in that case,"t" is small.

Is this justified?see,there is a delta function...

[Qfwfq]

You should make an effort to be clearer Kolahal, I came back to here to read your query again and I see your point a bit better, anyway you're certainly right that one can't just carelessly apply a steady-state law to the case of a point charge with velocity v. My memories from university courses are getting foggy in the details but you can certainly find good textbooks that treat these things properly. Of course, you could also try your hand at applying the Maxwell equations in covariant form but I don't think it's all that trivial to do.

[kolahal_b]

Well,I thank you very much.However,I had gone through Griffiths and Jackson abit.But,none of them treated this topic properly.