Calculate the Electric and Magnetic field for this Multipole Radiation

  • #1
Lambda96
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Homework Statement
Calculate the electric and magnetic field
Relevant Equations
none
Hi

I have problems with the following task

Bildschirmfoto 2024-11-23 um 14.44.43.png

I now wanted to try to calculate the vector potential, which according to my professor's script is defined as follows:

$$\mathbf{A}(\mathbf{x}) = \frac{1}{c} \int d^3\mathbf{x}' \frac{\mathbf{j}(\mathbf{x}')}{|\mathbf{x} - \mathbf{x}'|}$$

I have now visualized the problem as follows in 2D with regard to the x and z axes

Bildschirmfoto 2024-11-23 um 14.46.27.png

I'm not so sure, but does the electric as well as the magnetic field consist of a superposition when the particle is at the position ##z_1=a## and ##z_2=-a## and would look like ##E=E_1+E_2## and ##B=B_1+B_2##?

Then the following ##\mathbf{x'}_1=\left(\begin{array}{c} 0 \\\ 0 \\ a \end{array}\right)## and ##\mathbf{x'}_2=\left(\begin{array}{c} 0 \\\ 0 \\ -a \end{array}\right)## would hold and thus ##| \mathbf{x} -\mathbf{x'}_1 |=\sqrt{x^2+y^2+(z-a)^2}##

The electrical current density is defined as follows ##\mathbf{j}=\rho \mathbf{v}## and for this task:

$$\mathbf{j} = Q \cos(\omega t) \left[\delta(z - a) - \delta(z + a)\right] \delta(x) \delta(y) \cdot v \mathbf{e}_z$$

The vector potential can then be calculated as follows:

$$\mathbf{A}(\mathbf{x}) = \frac{1}{c} \int_{-\infty}^{\infty} d^3\mathbf{x}' \frac{Q \cos(\omega t) \left[\delta(z - a) - \delta(z + a)\right] \delta(x) \delta(y)}{\sqrt{x^2+y^2+(z-a)^2}}\cdot v \mathbf{e}_z$$

Would now simplify the potential because ##\mathbf{x'}_1=\left(\begin{array}{c} 0 \\\ 0 \\ a \end{array}\right)## as follows

$$\mathbf{A}(\mathbf{x}) = \frac{1}{c} \int_{-\infty}^{\infty} d\mathbf{z}' \frac{Q \cos(\omega t) \left[\delta(z - a) - \delta(z + a)\right] \delta(x) \delta(y)}{\sqrt{x^2+y^2+(z-a)^2}}\cdot v \mathbf{e}_z$$
 
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  • #2
I've just noticed, shouldn't it be?

$$\mathbf{A}(\mathbf{x}) = \frac{1}{c} \int_{-\infty}^{\infty} d\mathbf{z}' \frac{Q \cos(\omega t) \left[\delta(z - a) - \delta(z + a)\right]}{z-a}\cdot v \mathbf{e}_z$$
 
  • #3
Lambda96 said:
I'm not so sure, but does the electric as well as the magnetic field consist of a superposition when the particle is at the position ##z_1=a## and ##z_2=-a## and would look like ##E=E_1+E_2## and ##B=B_1+B_2##?
The charge accumulations ##q(t)## at ##z = a## and ##z = -a## are not due to a single particle moving back and forth. Instead, there is a time-dependent current ##I(t)## between the points ##z = -a## and ##z = a##.

1732470410086.png



The current ##I(t)## for ##|z|<a## does not depend on ##z## since otherwise there would be charge accumulations at points between ##z = -a## and ##z = a##.

So, the current is ##I = I(t)## for ##|z| < a## and ##I = 0## for ##|z| > a##. Use the Heaviside step function ##H## to express the current for all ##z## as $$I(z, t) = I(t) \left[ H(z-a) - H(z+a) \right].$$ Edit: The right side should read ##I(t) \left[ H(z+a) - H(z-a) \right]##. See post #5.

Note that ##I(t) = \dot q(t) = \dfrac {d }{dt} Q \cos(\omega t) = -\omega Q \sin \omega t##.

The current density vector is then $$\mathbf{j}(z,t) = I(z, t) \delta(x) \delta(y) \hat z = -\omega Q \sin \omega t\left[ H(z-a) - H(z+a) \right] \delta(x) \delta(y) \hat z.$$Edit: The right side should have ##\left[ H(z+a) - H(z-a) \right]## instead of ##\left[ H(z-a) - H(z+a) \right]##.

Verify that this expression for ##\mathbf j## satisfies the charge continuity equation $$\nabla \cdot \mathbf j = - \frac{\partial \rho(t, \mathbf x)}{\partial t}.$$ Recall that the derivative of the step function is a delta function: ##\dfrac{dH(z)}{dz} = \delta(z)##.
 
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  • #4
Thank you TSny for your help and explanation 👍👍

I have tried to calculate ##\nabla \cdot \mathbf{j}=-\frac{\partial \rho(t, \mathbf{x})}{\partial t}## and get the same result with a sign error. Have I done something wrong in the calculation?


$$\nabla \cdot \mathbf{j}=-\frac{\partial \rho(t, \mathbf{x})}{\partial t}$$
$$- \omega Q \sin(\omega) \left[ \delta(z-a)-\delta(z+a) \right] \delta(x)\delta(y)=\omega Q \sin(\omega) \left[ \delta(z-a)-\delta(z+a) \right] \delta(x)\delta(y)$$

I have now tried to calculate the potential with ##\mathbf{j}(z,t) = -\omega Q \sin \omega t\left[ H(z-a) - H(z+a) \right] \delta(x) \delta(y) \hat z## with the formula from post 2

$$\mathbf{A}(\mathbf{x}) = \frac{1}{c} \int_{-\infty}^{\infty} d\mathbf{z}' \frac{-\omega Q \sin \omega t\left[ H(z-a) - H(z+a) \right]}{z-a}$$

Unfortunately, the integral cannot be solved, which is why I assume that I have done something wrong :smile:
 
  • #5
Lambda96 said:
Thank you TSny for your help and explanation 👍👍

I have tried to calculate ##\nabla \cdot \mathbf{j}=-\frac{\partial \rho(t, \mathbf{x})}{\partial t}## and get the same result with a sign error. Have I done something wrong in the calculation?
The error is mine.

Instead of ##I(z, t) = I(t) \left[ H(z-a) - H(z+a) \right]##, it should be ##I(z, t) = I(t) \left[ H(z+a) - H(z-a) \right] ##.

Check that $$H(z+a) - H(z-a) = \begin{cases}
1 &\text{if } |z| < a \\
0 &\text{if } |z|>a\\
\end{cases}$$ Thus, $$\mathbf{j}(z,t) =-\omega Q \sin \omega t\left[ H(z+a) - H(z-a) \right] \delta(x) \delta(y) \, \hat {\mathbf z}.$$ Or,
$$\mathbf{j}(t)= \begin{cases}
-\omega Q \sin \omega t \delta(x) \delta(y) \hat {\mathbf z} &\text{if } |z| < a \\
\,\,\,\,0 &\text{if } |z|>a\\
\end{cases}$$

Lambda96 said:
I have now tried to calculate the potential with ##\mathbf{j}(z,t) = -\omega Q \sin \omega t\left[ H(z-a) - H(z+a) \right] \delta(x) \delta(y) \hat z## with the formula from post 2

$$\mathbf{A}(\mathbf{x}) = \frac{1}{c} \int_{-\infty}^{\infty} d\mathbf{z}' \frac{-\omega Q \sin \omega t\left[ H(z-a) - H(z+a) \right]}{z-a}$$

Unfortunately, the integral cannot be solved, which is why I assume that I have done something wrong :smile:

The formula for ##\mathbf{A}## is $$\mathbf{A}(\mathbf{x}, t) = \frac{1}{c} \int d^3\mathbf{x}' \frac{\mathbf{j}(\mathbf{x}', t -|\mathbf{x} - \mathbf{x}'|/c)}{|\mathbf{x} - \mathbf{x}'|}$$ Show that this reduces to $$\mathbf{A}(\mathbf{x}, t) = \frac{1}{c} \int_{-a}^a dz' \frac{I(t -|\mathbf{x} - z' \hat{\mathbf z}|/c)}{|\mathbf{x} - z' \hat{\mathbf z}|} \, \hat {\mathbf{z}}$$ You are only interested in the far zone where ##|\mathbf{x}| >> a##. So you can make appropriate approximations for ##|\mathbf{x} - z' \hat{\mathbf z}|## in the integrand. Hopefully, you've seen this type of approximation before.
 
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  • #6
Thank you for your help and the explanation TSny 👍👍

I have now started to show the following:
$$\mathbf{A}(\mathbf{x}, t) = \frac{1}{c} \int d^3\mathbf{x}' \frac{\mathbf{j}(\mathbf{x}', t -|\mathbf{x} - \mathbf{x}'|/c)}{|\mathbf{x} - \mathbf{x}'|} \quad \rightarrow \quad \mathbf{A}(\mathbf{x}, t) = \frac{1}{c} \int_{-a}^a dz' \frac{I(t -|\mathbf{x} - z' \hat{\mathbf z}|/c)}{|\mathbf{x} - z' \hat{\mathbf z}|} \, \hat {\mathbf{z}}$$

Unfortunately, I don't quite understand why ##t \quad \rightarrow \quad \frac{t -|\mathbf{x} - z' \hat{\mathbf z}|}{c}## holds, so I have now left the t as it is

$$\mathbf{A}(\mathbf{x}, t) = \frac{1}{c} \int d^3\mathbf{x}' \frac{\mathbf{j}(\mathbf{x}', t )}{|\mathbf{x} - \mathbf{x}'|}$$
$$\mathbf{A}(\mathbf{x}, t) = \frac{1}{c} \int d^3\mathbf{x}' \frac{-w Q \sin(\omega t) \left[H(z+a)-H(z-a)\right]\delta(x')\delta(y') \mathbf{\hat z}}{|\mathbf{x} - \mathbf{x}'|}$$
$$\mathbf{A}(\mathbf{x}, t) = \frac{1}{c} \int d^3\mathbf{x}' \frac{-w Q \sin(\omega t) \left[H(z+a)-H(z-a)\right]\delta(x')\delta(y') \mathbf{\hat z}}{\sqrt{(x-x')^2+(y-y')^2+(z-z')^2}}$$
$$\mathbf{A}(\mathbf{x}, t) = \frac{1}{c} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} dx' dy' dz'\frac{-w Q \sin(\omega t) \left[H(z+a)-H(z-a)\right]\delta(x')\delta(y') \mathbf{\hat z}}{\sqrt{(x-x')^2+(y-y')^2+(z-z')^2}}$$
$$\mathbf{A}(\mathbf{x}, t) = \frac{1}{c} \int_{- \infty}^{\infty} dz'\frac{-w Q \sin(\omega t) \left[H(z+a)-H(z-a)\right] \mathbf{\hat z}}{\sqrt{x^2+y^2+(z-z')^2}}$$
$$\mathbf{A}(\mathbf{x}, t) = \frac{1}{c} \int_{-a}^{a} dz'\frac{-w Q \sin(\omega t) \mathbf{\hat z}}{\sqrt{x^2+y^2+(z-z')^2}}$$
$$\mathbf{A}(\mathbf{x}, t) = \frac{1}{c} \int_{-a}^{a} dz'\frac{-w Q \sin(\omega t) \mathbf{\hat z}}{|\mathbf{x} - z' \hat{\mathbf z}|}$$
$$\mathbf{A}(\mathbf{x}, t) = \frac{1}{c} \int_{-a}^{a} dz'\frac{I(t) \mathbf{\hat z}}{|\mathbf{x} - z' \hat{\mathbf z}|}$$

Would that be correct?
 
  • #7
Lambda96 said:
I have now started to show the following:
$$\mathbf{A}(\mathbf{x}, t) = \frac{1}{c} \int d^3\mathbf{x}' \frac{\mathbf{j}(\mathbf{x}', t -|\mathbf{x} - \mathbf{x}'|/c)}{|\mathbf{x} - \mathbf{x}'|} \quad \rightarrow \quad \mathbf{A}(\mathbf{x}, t) = \frac{1}{c} \int_{-a}^a dz' \frac{I(t -|\mathbf{x} - z' \hat{\mathbf z}|/c)}{|\mathbf{x} - z' \hat{\mathbf z}|} \, \hat {\mathbf{z}}$$

Unfortunately, I don't quite understand why ##t \quad \rightarrow \quad \frac{t -|\mathbf{x} - z' \hat{\mathbf z}|}{c}## holds, so I have now left the t as it is
The time argument for ##\mathbf{j}## in the integrand needs to be the "retarded time" ##\dfrac{t -|\mathbf{x} - z' \hat{\mathbf z}|}{c}##. A change in current density at time ##t'## at point ##z'## in the antenna is not "felt" at the distant field point ##\mathbf x## until the later time ##t = t' + \dfrac{|\mathbf{x} - z' \hat{\mathbf z}|}{c}##. The time delay ##\dfrac{|\mathbf{x} - z' \hat{\mathbf z}|}{c}## is the "propagation time" for a signal traveling at light speed to get from ##z' \hat{\mathbf z}## to ##\mathbf{x}##.

In other words, ##\mathbf{A}(\mathbf x, t)## depends on what the source at ##z'## was doing at the earlier ("retarded") time ##t '= t - \dfrac{|\mathbf{x} - z' \hat{\mathbf z}|}{c}##.

Since different points of the antenna have different values of ##z'##, the retarded time for a fixed field-point-time ##t## is a function of ##z'##. This is the main complication in doing the integration.

Lambda96 said:
$$\mathbf{A}(\mathbf{x}, t) = \frac{1}{c} \int d^3\mathbf{x}' \frac{\mathbf{j}(\mathbf{x}', t )}{|\mathbf{x} - \mathbf{x}'|}$$
$$\mathbf{A}(\mathbf{x}, t) = \frac{1}{c} \int d^3\mathbf{x}' \frac{-w Q \sin(\omega t) \left[H(z+a)-H(z-a)\right]\delta(x')\delta(y') \mathbf{\hat z}}{|\mathbf{x} - \mathbf{x}'|}$$
$$\mathbf{A}(\mathbf{x}, t) = \frac{1}{c} \int d^3\mathbf{x}' \frac{-w Q \sin(\omega t) \left[H(z+a)-H(z-a)\right]\delta(x')\delta(y') \mathbf{\hat z}}{\sqrt{(x-x')^2+(y-y')^2+(z-z')^2}}$$
$$\mathbf{A}(\mathbf{x}, t) = \frac{1}{c} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} dx' dy' dz'\frac{-w Q \sin(\omega t) \left[H(z+a)-H(z-a)\right]\delta(x')\delta(y') \mathbf{\hat z}}{\sqrt{(x-x')^2+(y-y')^2+(z-z')^2}}$$
$$\mathbf{A}(\mathbf{x}, t) = \frac{1}{c} \int_{- \infty}^{\infty} dz'\frac{-w Q \sin(\omega t) \left[H(z+a)-H(z-a)\right] \mathbf{\hat z}}{\sqrt{x^2+y^2+(z-z')^2}}$$
$$\mathbf{A}(\mathbf{x}, t) = \frac{1}{c} \int_{-a}^{a} dz'\frac{-w Q \sin(\omega t) \mathbf{\hat z}}{\sqrt{x^2+y^2+(z-z')^2}}$$
$$\mathbf{A}(\mathbf{x}, t) = \frac{1}{c} \int_{-a}^{a} dz'\frac{-w Q \sin(\omega t) \mathbf{\hat z}}{|\mathbf{x} - z' \hat{\mathbf z}|}$$
$$\mathbf{A}(\mathbf{x}, t) = \frac{1}{c} \int_{-a}^{a} dz'\frac{I(t) \mathbf{\hat z}}{|\mathbf{x} - z' \hat{\mathbf z}|}$$

Would that be correct?

Yes, that all looks good except for the need to use the retarded time in the integrand.
 
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  • #8
Thank you TSny for your help and the explanation regarding the retarded time 👍👍, I have now inserted this into my expression :smile:

I've been thinking about the approximation and unfortunately I can't get any further. The only thing I can think of would be a binomial expansion, as I often had this in my calculus course, so

$$|\mathbf{x}-z' \mathbf{\hat z}| \approx |\mathbf{x}| \Bigl( 1- \frac{z z'}{|\mathbf{x}|^2} \Bigr)$$

Did you mean this approximation?
 
  • #9
Lambda96 said:
I've been thinking about the approximation and unfortunately I can't get any further. The only thing I can think of would be a binomial expansion, as I often had this in my calculus course, so

$$|\mathbf{x}-z' \mathbf{\hat z}| \approx |\mathbf{x}| \Bigl( 1- \frac{z z'}{|\mathbf{x}|^2} \Bigr)$$

Did you mean this approximation?
Yes, that's it.

If we let ##r = |\mathbf x##|, then your result can be expressed as $$|\mathbf{x}-z' \mathbf{\hat z}| \approx r ( 1- \frac{z z'}{r^2} )$$ A nice way to express this is to note that ##\dfrac {z}{r} = \cos\theta##, where ##\theta## is the angle between the z-axis and ##\mathbf x##. Thus, $$|\mathbf{x}-z' \mathbf{\hat z}| \approx r ( 1- \frac{z' \cos \theta}{r} ) = r - z' \cos \theta$$
Note that ##|\mathbf{x}-z' \mathbf{\hat z}|## occurs in two places in the integrand for ##\mathbf A(\mathbf x, t)##: (1) in the denominator and (2) in the retarded time for the argument of ##\mathbf j##. In the far zone, we only need to keep terms where ##|\mathbf A|## falls off with distance ##r## as ##1/r## and we can ignore terms for which ##|\mathbf A|## falls off as ##1/r^2## or faster. This helps with making the approximation for the denominator. You'll also need to work on getting the correct approximation for the argument of ##\mathbf j##.
 
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  • #10
Thanks again for your help and explanations TSny 👍👍

I have now started with the denominator. I now have the expression ##|\mathbf{x}-z' \mathbf{\hat z}| \approx r ( 1- \frac{z' \cos \theta}{r} ) = r - z' \cos \theta## since we are interested in the far zone, the following applies ##r \gg z'## which means that I can simply ignore the term ##z' \cos \theta## and the integral looks like this:

$$\mathbf{A}(\mathbf{x}, t) = \frac{1}{c} \int_{-a}^{a} dz'\frac{- \omega Q \sin(\omega(t-\frac{|\mathbf{x}-z' \mathbf{\hat z}|}{c}))}{r}$$

Unfortunately I still have some problems with the numerator, I have now used the approximation in the sine, which then looks like this:

$$\omega \left( t - \frac{\lvert \mathbf{x} - z' \hat{\mathbf{z}} \rvert}{c} \right) \approx \omega \left( t - \frac{r }{c} + \frac{z' \cos\theta}{c} \right)=\omega \left( t - \frac{r}{c} \right) + \frac{\omega z' \cos\theta}{c}$$

If the ##\omega## didn't appear in ##\frac{\omega z' \cos\theta}{c}##, I would have simply argued that this term can simply be omitted, as it is extremely small, because ##\omega## could be a large number, which is why it no longer works. Then I wanted to use ##\sin(A+B)=\cos(B)\sin(A)+\cos(A)\sin(B)##, but that doesn't get rid of the term ##\sin(\cos(\theta))##, it actually makes it worse 🙃
 
  • #11
Lambda96 said:
I have now started with the denominator. I now have the expression ##|\mathbf{x}-z' \mathbf{\hat z}| \approx r ( 1- \frac{z' \cos \theta}{r} ) = r - z' \cos \theta## since we are interested in the far zone, the following applies ##r \gg z'## which means that I can simply ignore the term ##z' \cos \theta## and the integral looks like this:

$$\mathbf{A}(\mathbf{x}, t) = \frac{1}{c} \int_{-a}^{a} dz'\frac{- \omega Q \sin(\omega(t-\frac{|\mathbf{x}-z' \mathbf{\hat z}|}{c}))}{r}$$
Looks good.

Lambda96 said:
Unfortunately I still have some problems with the numerator, I have now used the approximation in the sine, which then looks like this:

$$\omega \left( t - \frac{\lvert \mathbf{x} - z' \hat{\mathbf{z}} \rvert}{c} \right) \approx \omega \left( t - \frac{r }{c} + \frac{z' \cos\theta}{c} \right)=\omega \left( t - \frac{r}{c} \right) + \frac{\omega z' \cos\theta}{c}$$
Good.

Lambda96 said:
If the ##\omega## didn't appear in ##\frac{\omega z' \cos\theta}{c}##, I would have simply argued that this term can simply be omitted, as it is extremely small, because ##\omega## could be a large number, which is why it no longer works.
Right. You need to keep ##\frac{\omega z' \cos\theta}{c}##. Very good.

Lambda96 said:
Then I wanted to use ##\sin(A+B)=\cos(B)\sin(A)+\cos(A)\sin(B)##,
Good.

Lambda96 said:
but that doesn't get rid of the term ##\sin(\cos(\theta))##, it actually makes it worse 🙃
When integrating over ##z'##, ##\theta## is fixed.
Since you are integrating from ##-a## to ##a##, consider if ##\sin(\frac{\omega z' \cos\theta}{c})## and ##\cos(\frac{\omega z' \cos\theta}{c})## are even or odd functions of ##z'##.
 
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  • #12
Thanks again for your help TSny and the tip 👍👍 that ##\cos## and ##\sin## are even and odd functions.

If I have not miscalculated, I have got the following result:

$$\mathbf{A}(r,t)=\frac{-2Q \omega \sin\Bigl(\frac{w z' \cos(\theta)}{c} \Bigr)\sin\Bigl(\omega \Bigl( \frac{t-r}{c}\Bigr)\Bigr)}{r \cos(\theta)}$$
 
  • #13
Lambda96 said:
If I have not miscalculated, I have got the following result:

$$\mathbf{A}(r,t)=\frac{-2Q \omega \sin\Bigl(\frac{w z' \cos(\theta)}{c} \Bigr)\sin\Bigl(\omega \Bigl( \frac{t-r}{c}\Bigr)\Bigr)}{r \cos(\theta)}$$
Getting close. But, I see some typos and other errors:

(1) ##\mathbf{A}(r,t)## should be ##\mathbf{A}(r, \theta, t)##.

(2) You need a unit vector to specify the direction of ##\mathbf {A}##.

(3) Your expression does not have the correct dimensions for ##\mathbf{A}##.

(4) The argument for the second sine function has a typo.

(5) Your result should not depend on ##z'##. You integrated over ##z'## (hopefully).
 
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  • #14
Thanks again for your help TSny 👍

I have now corrected all the points you listed, except for point (3) where I am not sure if the expression now has the correct dimension:

$$\mathbf{A}(r,t, \theta)=\frac{-2Q \omega \sin\Bigl(\frac{w a \cos(\theta)}{c} \Bigr)\sin\Bigl(\omega \Bigl( \frac{ct-r}{c}\Bigr)\Bigr)}{r \cos(\theta)} \mathbf{\hat z}$$
 
  • #15
Lambda96 said:
I have now corrected all the points you listed, except for point (3) where I am not sure if the expression now has the correct dimension:

$$\mathbf{A}(r,t, \theta)=\frac{-2Q \omega \sin\Bigl(\frac{w a \cos(\theta)}{c} \Bigr)\sin\Bigl(\omega \Bigl( \frac{ct-r}{c}\Bigr)\Bigr)}{r \cos(\theta)} \mathbf{\hat z}$$
Very close.

Regarding the dimensions, pick a formula that makes it easy to see what the dimensions should be. For example $$\mathbf{A}(\mathbf{x}) = \frac{1}{c} \int d^3\mathbf{x}' \frac{\mathbf{j}(\mathbf{x}')}{|\mathbf{x} - \mathbf{x}'|}.$$
 
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  • #16
Tanke for your help TSny 👍

The individual dimensions (SI) of the terms are as follows:

$$c=\frac{m}{s}, \quad |\mathbf{x} - \mathbf{x}'|=m, \quad \mathbf{j}=\frac{A}{m^2}$$

If I now insert this into the equation from your post 15, I get ##\frac{s}{m} \frac{\frac{A}{m^2}}{m}=\frac{A s}{m^4}##, but if I'm not mistaken I should get ##\frac{V s}{m}##.
 
  • #17
We are working in Gaussian units where $$\mathbf{A}(\mathbf{x}) = \frac{1}{c} \int d^3\mathbf{x}' \frac{\mathbf{j}(\mathbf{x}')}{|\mathbf{x} - \mathbf{x}'|}.$$ Consider dimensions rather than units. Using square brackets for indicating the dimension of a quantity, $$[\mathbf A] = \frac{1}{[c]} [d^3x] [\mathbf j] \frac 1 {[x]} = \frac{1}{[L]/[T]} [L]^3 [\mathbf j] \frac 1 {[L]}$$ where ##[L]## is dimension of length and ##[T]## is dimension of time. From the definition of current density ##\mathbf j##, work out ##[\mathbf j]## in terms of ##[Q], [L],## and ##[T]##. Here, ##[Q]## is the dimension of charge. Thus, reduce ##[\mathbf A##] to some combination of ##[Q], [L]##, and ##[T]##.

Compare with the dimensions of your result for ##\mathbf A## in post ##14##.
 
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  • #18
Lambda96 said:
The individual dimensions (SI) of the terms are as follows:
$$c=\frac{m}{s}, \quad |\mathbf{x} - \mathbf{x}'|=m, \quad \mathbf{j}=\frac{A}{m^2}$$
Don't forget to include the ##d^3\mathbf{x}'=m^3## term!
 
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  • #19
Thanks again for all your help and explanations TSny 👍👍 also thanks renormalize for your help 👍

Now I see where I made the mistake :smile:
 
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  • #20
Lambda96 said:
Thanks again for all your help and explanations TSny 👍👍 also thanks renormalize for your help 👍

Now I see where I made the mistake :smile:
Only in PF you get (re)normalized help!
 
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