Engineering Calculating the flux through a certain radius (concentric charged spheres)

[falyusuf]

Homework Statement

Attached below.

Relevant Equations

Attached below.

Question

My attempt for the 1st part, when r = 1.5 m

Could someone confirm my answer?

 

[BvU]

Hi,

Do you know what a shell is ?
You seem to integrate something from 1 to 1.5 m. What is that ?
Did you notice the dimension of the given charge distributions ? Does that match your $\rho_v$ ?
I also have trouble understanding what your symbols mean. $\phi = Q$ ?

$\$

 

[falyusuf]

You seem to integrate something from 1 to 1.5 m. What is that ?

also have trouble understanding what your symbols mean. ϕ=Q ?

 

[BvU]

Sparse with words we are, eh ?
So what is flux in your context ? Units ?
You are asked to calculate the flux through $r = 1.5$ m as a starter in part (a).

My attempt for the 1st part, when r = 1.5 m

In post #3 the volume integral you wrote in post #1 is replaced by a surface integral. Good ! And you write $$\phi = \oint \overline D\, \overline{dS} = Q_{enc} =\int \rho_s ds$$ where I must suppose $D = \varepsilon_0 E$ ?

In my book the electric flux in such a nice spherically symmetric case is $\Phi = EA$ with E the electric field strength in Newton/Coulomb and A the area in m² . In other words, I get $$\Phi = \oint {\overline {D}\over \varepsilon_0}\; \overline{dS} = {Q_{enc}\over \varepsilon_0}$$ and the units are Nm²/C.

We are still in part (a) first question. I agree that $Q_{enc} = \rho_s \, 4\pi \,r^2$ with $r = 1 m$.
second and third (a) questions remain unanswered ?
---

Moving on to part (b) first question:

Your exercise composer makes life difficult by re-using symbol $r$ in r = 1.5 m where $D$ is asked for. We no longer need the $\varepsilon_0$.
You are smart enough to replace $r$ by $R$ (without explaining) but then you take $R= 1$ m (given) ?

$\$

 

[falyusuf]

Sparse with words we are, eh ?
So what is flux in your context ? Units ?
You are asked to calculate the flux through $r = 1.5$ m as a starter in part (a).

In post #3 the volume integral you wrote in post #1 is replaced by a surface integral. Good ! And you write $$\phi = \oint \overline D\, \overline{dS} = Q_{enc} =\int \rho_s ds$$ where I must suppose $D = \varepsilon_0 E$ ?

In my book the electric flux in such a nice spherically symmetric case is $\Phi = EA$ with E the electric field strength in Newton/Coulomb and A the area in m² . In other words, I get $$\Phi = \oint {\overline {D}\over \varepsilon_0}\; \overline{dS} = {Q_{enc}\over \varepsilon_0}$$ and the units are Nm²/C.

We are still in part (a) first question. I agree that $Q_{enc} = \rho_s \, 4\pi \,r^2$ with $r = 1 m$.
second and third (a) questions remain unanswered ?
---

Moving on to part (b) first question:

Your exercise composer makes life difficult by re-using symbol $r$ in r = 1.5 m where $D$ is asked for. We no longer need the $\varepsilon_0$.
You are smart enough to replace $r$ by $R$ (without explaining) but then you take $R= 1$ m (given) ?

$\$

I was so confused in using many r's, so I use different symbols, and tried to solve it again. Here's my attempt with my explanation:

And regarding the following formula:

This is given in the textbook.

 

[Babadag]

If the flux it is the total inside charge you have to calculate D at r=2.5 [or r=0.5] and not at r=1.5