# Lienard-Wiechert Potentials Issue

I have had some difficutlies in trying to obtain the Lienard-Wiechert potentials of a moving "point charge" and would greatly appreciate any help in this matter. I will adopt the definition of "point charge" as a "sufficiently small charge", meaning a charge whose dimensions are at all times negligible compared to the magnitude of its position vector.

I should also stress out first that the following arguments are made outside the scope of Special Relativity as the LW theory was developed before 1905.

First, I will begin by discussing how the velocity vector of a moving objects appears to be changing when viewed from different positions. I will make use of the relation: $$t' = t - \frac{r}{c}$$ where $$t'$$ is the retarded time, $$t$$ is the time as measured at the observation point (OP) and $$r$$ is the retarded position of the object. Let's call "proper velocity" the quantity: $$\dot{\mathbf{r'}} = \frac{d\mathbf{r}}{dt'}$$ However, the apparent velocity of the object as measured at the OP will be: $$\dot{\mathbf{r}} = \frac{d\mathbf{r}}{dt}$$ and since $$\frac{dt}{dt'} = 1+ \frac{\frac{dr}{dt'}}{c}$$ we get: $$\dot{\mathbf{r}} = \frac{\dot{\mathbf{r'}}}{1 + \frac{\dot{r'}}{c}}$$ So the object appears to be moving faster when its velocity vector points towards the OP (and of course slower when it moves away from the OP). Please note that above: $$\dot{r'} = \mathbf{r}\cdot\dot{\mathbf{r'}}$$ all bold terms being vectors throughout this post.

Secondly, I want to discuss how the apparent size of a moving object changes as seen from different OPs around the object. To make the issue simpler I will restrict the discussion to the one dimensional case. Let's assume that an object of length $$L'$$ is positioned at some distance away from the OP and then suddently begins to move towards towards the OP with proper velocity $$\dot{\mathbf{r'}}$$. As the front of the object is closer than its back side with respect to the OP, it will appear that the front side will begin to move first and only after a time interval $$\Delta{t} = \frac{L'}{c}$$ will the back side begin to move. During this time interval the object will seem to be "dilating" to an apparent length of: $$L = L'(1-\frac{\dot{r}}{c})$$ (with $$\dot{r} = \mathbf{r}\cdot\dot{\mathbf{r}}$$) or, if we express the apparent length in terms of the "proper velocity": $$L = \frac{L'}{1+\frac{\dot{r'}}{c}}$$

Now, if the object is electrically charged, then its apparent charge will increase proportionally with its apparent volume and we get the usual term of $$\frac{1}{1+\frac{\dot{r'}}{c}}$$ in the LW potentials for the scalar potential:

$$\Phi = \frac{q}{4\pi\epsilon_0 r(1+\frac{\dot{r'}}{c})}$$ or expressed in terms of the apparent velocity:$$\Phi = \frac{q(1-\frac{\dot{r}}{c})}{4\pi\epsilon_0r}$$

So far so good. Now here comes my difficulty. It seems reasonable to me that the vector potential, being a function of both the charge and velocity of the particle, should be written in terms of the apparent charge and apparent velocity of the particle at the OP:

$$\mathbf{A} = \frac{q(1-\frac{\dot{r}}{c})\mathbf{\dot{r}}}{4\pi\epsilon_0c^2r}$$ If I try to express it however in terms of the "proper veolcity" of the particle I get: $$\mathbf{A} = \frac{q\mathbf{\dot{r'}}}{4\pi\epsilon_0c^2r\left(1+\frac{\dot{r'}}{c}\right)^2}$$ so an extra factor of $$\frac{1}{(1+\frac{\dot{r'}}{c})}$$ in the denominator compared to the default LW expressions for the vector potential! I clearly must be doing something wrong here but it's not obvious to me what that is...

It is interesing to note that both the scalar and vector potentials when expressed in terms of the apparent charge and velocity, i.e.: $$\Phi = \frac{q(1-\frac{\dot{r}}{c})}{4\pi\epsilon_0r}$$ and $$\mathbf{A} = \frac{q(1-\frac{\dot{r}}{c})\mathbf{\dot{r}}}{4\pi\epsilon_0c^2r}$$ seem to obey Lorentz's gauge since $$\mathbf{A} = \Phi\mathbf{\dot{r}}/c^2$$.

Any feedback regarding my error/errors in this post will be greatly appreciated.

• Your concept of proper velocity seems very strange. Objects have no velocity in their own reference frame. Commented Aug 29, 2023 at 16:59
• Why not use Wikipedia’s derivation, or the one in your textbook? Commented Aug 29, 2023 at 17:01
• I guess I should have made it a bit more clear what I mean by "proper velocity". What I mean by this expression is the particle's velocity when measured at an observation point where the radial velocity component of the paricle is zero, in other words the particle's velocity vector is perpendicular to its position vector at this OP. Commented Aug 29, 2023 at 17:33
• I had to edit the post quite a few times to make the argumet and some formulas clearer or to correct some obvious misspells. Ghoster comment about my proper velocity definition is justified as it was ill defined initially in the post. Commented Aug 30, 2023 at 10:04
• My textbooks (D. Griffiths - "Introduction to Electrodynamics" and "The Feynman Lectures on Physics") seem to only account for the modification of the particle's charge at the OP in deriving the LW potentials but don't seem to account for the modification velocity of the particle (from "proper velocity" to the "apparent velocity" in the context of this post). I am not claming by any means that they are wrong and I am right, I would just like to receive some feedback on why using the "apparent velocity" of the point charge at the OP is wrong. Commented Aug 30, 2023 at 13:06

I believe your difficulties come from your idea of "apparent velocity".

LW gives a prediction of the potential created by a charge at some position $$r$$ on a charge at position $$OP$$. The charge was at position $$r$$ at time t'. If you choose a different $$OP$$ with a different $$|r-OP|$$, then the charge could be at some other location with some other velocity and other acceleration. It might just as well be a different charge.

For LW purposes, all we care about is the proper velocity. It's what we use to calculate potential. Don't worry that you get a different result when you calculate with apparent velocity. That result isn't the one you want.

You try to calculate $$A$$ starting from apparent position and apparent velocity. If you accept it as defined from proper position and proper velocity, then it works.

• J Thomas - I agree with you that my difficulty is caused by the use of the "apparent velocity" rather than the "proper velocity". However, what seems to me inconsistent is that we're already using the "apparent charge" instead of the "proper charge" (the value of the charge in a static reference frame) in the standard LW potentials but, for some reason that eludes me, we're not using the "apparent velocity" for the potentials at the OP. So, in conclusion, why are we only accounting for the modification of the electric charge but not for the velocity of the particle at the OP? Commented Aug 31, 2023 at 13:16
• I don't understand. It looks to me like you're using the same $q$. Commented Aug 31, 2023 at 18:01
• In the context of the post, the product of $q$ and the $\frac{1}{(1+\frac{\dot{r'}}{c})}$ term is the apparent charge as measured at the OP. Obviously, this quantity is not a constant (as in its static frame) but a velocity dependent parameter so it must be express as such. Feynman and Griffiths account for this effect (although they don't use this terminology) as a result of the dilation or contraction of the size of a small electrically charged object when its velocity points towards or away from the OP. Commented Aug 31, 2023 at 19:03
• You're coming up with a way to look at the equation that makes sense to you. I can't say you're wrong if that's how it makes sense. The way it looks to me, you can have a point particle and at one moment it initiates a potential that spreads in a sphere at lightspeed. A few moments later it initiates another potential that spreads in a sphere at lightspeed. If the center of the second sphere is different, they will be squeezed together in one direction and spread in the other, like doppler. No need to think the point particle's size has changed. Commented Sep 1, 2023 at 1:56
• What I understand from your description above is an effect (much like Doppler) which would just account for the particle's velocity observed at the OP as "apparent velocity" instead of "proper velocity" (since after the particle initiates the first potential it moves to a different location before emitting the second potential after T seconds, so the time interval between the detection of the two potentials, T',would in general differ at the OP). I cannot (and this is my own shortcomming not yours) imagine how a dimensionless charge can produce the same effect as a non-dimensionless charge. Commented Sep 3, 2023 at 0:52