Find the net outward flux through the surface and the charge density

[vboyn12]

Homework Statement

Find the net outward flux through the surface and the charge density. (Help please)

Relevant Equations

Relevant equations in a photo below. I try to solve it but the result is 0 which makes me confused

 

[vboyn12]

 

[TSny]

When taking the divergence, note that the $\theta$ component of $\mathbf D$ has a numerical coefficient of 10, not 20.

After you find the charge density, you might be able to see whether or not a zero answer for the flux through the spherical surface makes sense.

What do you get for part (b)?

 

[TSny]

Sorry. I now see where the factor of 20 comes from in evaluating the $\theta$ component of the divergence. Your work looks OK to me.

 

[vboyn12]

When taking the divergence, note that the $\theta$ component of $\mathbf D$ has a numerical coefficient of 10, not 20.

After you find the charge density, you might be able to see whether or not a zero answer for the flux through the spherical surface makes sense.

What do you get for part (b)?

I think it must be 20 because when taking partial derivative of D(theta component)*sin(theta) respect to theta we can obtain derivative of sin(theta)^2=2sin(theta)cos(theta). This is the first time I post thread so excuse me about the math formulas

 

[vboyn12]

Sorry. I now see where the factor of 20 comes from in evaluating the $\theta$ component of the divergence. Your work looks OK to me

Sorry. I now see where the factor of 20 comes from in evaluating the $\theta$ component of the divergence. Your work looks OK to me.

thank you. Can you give me some hints to do part (b), please?

 

[TSny]

I think it must be 20 because when taking partial derivative of D(theta component)*sin(theta) respect to theta we can obtain derivative of sin(theta)^2=2sin(theta)cos(theta). This is the first time I post thread so excuse me about the math formulas

Yes, you are right. Your work looks correct.

 

[TSny]

thank you. Can you give me some hints to do part (b), please?

All you need is a minor modification of your work for part (a). Would any of the limits of integration change?

 

[vboyn12]

All you need is a minor modification of your work for part (a). Would any of the limits of integration change?

theta from 0 to pi/2, is it correct?

 

[TSny]

theta from 0 to pi/2, is it correct?

Yes. Do you know if the hemisphere is meant to include a flat base?

 

[vboyn12]

Yes. Do you know if the hemisphere is meant to include a flat base?

So we have to take a double integral of the flat base with limits r from 0 to 1 and phi from 0 to 2pi, i guest

 

[TSny]

So we have to take a double integral of the flat base with limits r from 0 to 1 and phi from 0 to 2pi, i guest

Yes. However, there could be a difficulty here due to the fact that the field blows up as $1/r^3$ for $r$ going to zero. So, maybe they don't want you to include the base. I don't know.

This singularity of the field also means that the divergence of $\mathbf D$ is not actually defined at $r = 0$. Your result $\nabla \cdot \mathbf D = 0$ is valid for all points except at the origin.

 

[vboyn12]

Yes. However, there could be a difficulty here due to the fact that the field blows up as $1/r^3$ for $r$ going to zero. So, maybe they don't want you to include the base. I don't know.

This singularity of the field also means that the divergence of $\mathbf D$ is not actually defined at $r = 0$. Your result $\nabla \cdot \mathbf D = 0$ is valid for all points except at the origin.

thank you a lot.

 

[vboyn12]

thank you a lot.

Can you take some time to have a look at another topic of mine?