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## Homework Statement

The electric flux density in free space produced by an oscillating electric charge placed at the origin is given by

[tex]\vec{D}=\hat{r}\frac{10^{-9}}{4\pi r^2}cos(wt-\beta r), \ \ where \ \beta=w \sqrt{\mu_0 \epsilon_0}[/tex]

Find the time-average charge that produces this electric flux density

## Homework Equations

[tex]div\vec{D}=\rho \ (1) \\ \int_S \vec{D}.\vec{ds}=Q_{int} \ (2)[/tex]

## The Attempt at a Solution

This is the exercise 1.9 from Balanis' Advanced Engineering Electromagnetics. As the charge is oscillating, we can see a propagation delay in the cosine argument. Applying the divergent in D in spherical coordinates, we get the value

[tex]div\vec{D}=\frac{1}{r^2}\frac{\partial}{\partial r}(r^2 cos(wt-\beta r)\frac{10^{-9}}{4\pi r^2}) = \beta sin(wt-\beta r)\frac{10^{-9}}{4\pi r^2}[/tex]

And that is the value of the charge density. As expected, it goes to infinity when r→0, but it isn't zero when r≠0. In the other hand, equation (2) gives the value of internal charge Q

_{int}=10

^{-9}cos(wt-βr), as expected from a oscillating punctual charge. Why equation (1) gives a wrong value?

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