Charge density from electric flux density

[freezer]

Homework Statement

Determine the charge density due to the following electric flux density:

$$\overrightarrow{D} = \hat{r}4rsin(\phi ) + \hat{\phi}2rcos(\phi)+\hat{z}2z^{2}$$

Homework Equations

$$\rho _{v} = \triangledown \cdot \vec{D}$$

$$\rho _{v} = \frac{\partial }{\partial r} + \frac{\partial }{\partial \phi}+ \frac{\partial }{\partial z}$$

The Attempt at a Solution

$$\rho _{v} = \frac{\partial }{\partial r} 4rsin(\phi) + \frac{1}{r}\frac{\partial }{\partial \phi}2rcos(\phi)+ \frac{\partial }{\partial z}2z^{2}$$

$$\rho _{v} = 4sin(\phi)- 2sin(\phi) + 4z$$

$$\rho _{v} = 2sin(\phi) + 4z$$

does this look correct?

 

[rude man]

Homework Statement

Determine the charge density due to the following electric flux density:

$$\overrightarrow{D} = \hat{r}4rsin(\phi ) + \hat{\phi}2rcos(\phi)+\hat{z}2z^{2}$$

Homework Equations

$$\rho _{v} = \triangledown \cdot \vec{D}$$

$$\rho _{v} = \frac{\partial }{\partial r} + \frac{\partial }{\partial \phi}+ \frac{\partial }{\partial z}$$

The Attempt at a Solution

$$\rho _{v} = \frac{\partial }{\partial r} 4rsin(\phi) + \frac{1}{r}\frac{\partial }{\partial \phi}2rcos(\phi)+ \frac{\partial }{\partial z}2z^{2}$$

does this look correct?

There is a term missing here.

 

[freezer]

There is a term missing here.

I am not seeing it. For cylindrical I am only seeing the 1/r on the phihat term

 

[rude man]

I am not seeing it. For cylindrical I am only seeing the 1/r on the phihat term

Stick a term 4sinø in there somewhere, anywhere. No partial derivative. Just that term. (It's part of the rhat coefficient: 4rsinø/r).

 

[rezeew]

I would say that your solution looks correct. However, it would be helpful to provide some context or explanation for your calculations and how you arrived at your answer. Additionally, you may want to check your units to make sure they are consistent with the given units for electric flux density and charge density.