Find charge distribution of a point charge at origin.

[yungman]

This is part of the problem in the exercise to find charge distribution of a point charge at origin. I know $$\nabla \cdot \vec E = \frac {\rho}{\epsilon_0}$$

$$\rho = \epislon \nabla \cdot \hat r \frac q {4\pi\epsilon_0 r^2} \;\hbox { where } \vec r = \hat x x + \hat y y + \hat z z$$

$$\nabla \cdot \vec E = \frac q {4\pi\epsilon_0}\left [ \frac {\partial }{\partial x} \left ( \frac x {r^3} \right ) + \frac {\partial }{\partial y} \left ( \frac y {r^3} \right ) + \frac {\partial }{\partial z} \left ( \frac z {r^3} \right ) \right ]$$

$$\frac {\partial }{\partial x} \left ( \frac x {r^3} \right ) = \frac { r^3 - x d(r^3)}{r^6} = \frac { r^3 - 6x^2 r^2}{r^6} = \frac 1 {r^3}-\frac {6x^2}{r^4}$$

The other two can be worked out as above for y and z.

$$\Rightarrow\; \nabla \cdot \vec E = \frac q {4\pi\epsilon_0} \left [ \frac 3 {r^3} - \frac {6 (x^2 + y^2 + z^2)}{r^4} \right ] = \frac q {4\pi\epsilon_0} \left [ \frac 3 {r^3} - \frac 6 {r^2} \right ] = \frac {\rho}{\epsilon_0}$$

$$\rho = \frac q {4\pi} \left [ \frac 3 {r^3} - \frac 6 {r^2} \right ]$$

The book gave:

$$\rho = q \delta^3(\vec r)$$

I don't know what is $\delta$ in the book!

Can anyone verify my work?

Thanks

 

[tiny-tim]

hi yungman!

sorry, that's completely wrong …

ρ is zero except at the origin, where it's infinite

δ is the Dirac delta function (strictly, a distribution rather than a function, since it only "lives" inside integrals), which you'll have to look up

 

[kloptok]

hi yungman!

both ρ and E are zero except at the origin, where they're infinite

$$\rho$$ is zero except at the origin, but E can't be. Surely it must be the E-field of a point charge, $$E=\frac{q}{4\pi \epsilon_0 r^2}$$ ?

 

[tiny-tim]

oops!

oops! I've no idea why i wrote that!

(i've edited it now)

thanks for the correction, kloptok ​

 

[yungman]

hi yungman!

sorry, that's completely wrong …

ρ is zero except at the origin, where it's infinite

δ is the Dirac delta function (strictly, a distribution rather than a function, since it only "lives" inside integrals), which you'll have to look up

Hi Tiny Tim

I don't know what I am thinking, I should know δ! And I should know that. This is a problem in Gauge Transformation and I never even stop and think! The problem started out as $$\vec A = \frac { qt}{4\pi\epsilon_0 r^2} \hat r$$

And asked to find E, B, J and current distribution. I just blindly go through the steps and solve the problem without stop and think! .

How can I avoid making the same mistake because without stop and think, I just apply the divergence and get the wrong answer!??

Thanks! Today is my 58th birthday and I sure acted stupidly!