Verifying Gauss' Law: Homework Statement Solutions

[ryukyu]

Homework Statement

For the given flux density: $$\vec{D}$$=(2y2+z)$$\widehat{x}$$+(4xy)$$\widehat{y}$$^+(xz)$$\widehat{z}$$
a)Determine the charge density.
b)Find the total charge enclosed if the surface is 0<x<1, 0<y<1, 0<z<1 (unit cube)
c)Confirm Gauss’s law by finding the net flux through the surface of the volume.

Homework Equations

The Attempt at a Solution

I used divergence to find the $$\rho$$_v=5x

To find Qenc I integrated $$\int\int\int$$5xdxdydz and came up with
Q_enc=5/2 C

The last step I know is to verify that $$\oint$$$$\vec{D}$$dS=Q_enc.

From what I gather since the divergence only has an x-component we will integrate only the x-component over the dxdydz, but this gives me 7/2. I'm guessing both are incorrect, but obviously at least one of them is.

 

[ehild]

1.

The last step I know is to verify that $$\oint$$$$\vec{D}$$dS=Q_enc.

From what I gather since the divergence only has an x-component we will integrate only the x-component over the dxdydz, but this gives me 7/2. I'm guessing both are incorrect, but obviously at least one of them is.

The divergence is a scalar, and you have to integrate the flux density for all sides of the cube. Remember that the surface element dS is a vector normal to the surface.


ehild

 

[ryukyu]

Thanks for the response and the insight.

So for the top of said cube I would integrate the z-hat coefficient by dydx?
the bottom by z-hat (-dydx)
the right by y-hat (dxdz)
left by y-hat(-dxdz)
front by x-hat(dydz)
and back x-hat(-dydz)...

 

[ehild]

It looks OK.

ehild

 

[ryukyu]

I apologize for not saying thanks again. I do find this site a valuable resource in attempting to learn this material instead of just blindly using equations and hoping that things work out.

 

[ehild]

You are welcome.

ehild