Electron's Velocity: Calculating w/ Gauss' Thm

[Kernul]

Homework Statement

Two thin cylindrical shells, respectively with radius $R_1 = 12 cm$ and $R_2 = 20 cm$ and height $D = 100 m$, are concentric and characterized by a charge density $\lambda_1 = -0.38 \frac{\mu C}{m}$, for the internal cylinder, and $\lambda_2 = 0.32 \frac{\mu C}{m}$, for the external one. An electron ($m_e = 9.1 \cdot 10^{-31} kg$ and a charge $e = 1.6 \cdot 10^{-19} C$) is released stationary from the internal cylinder's surface; what is the velocity when it reaches the other cylinder?

Homework Equations

Gauss Teorem
$\Phi_S (\vec E_0) = \int_S \vec E_0 \cdot d\vec S = \frac{Q}{\epsilon_0}$
and
$a = \frac{q}{m}E$

The Attempt at a Solution

So, I started by calculating the electric field on both the cylinders.
$\Phi_S (\vec E_1) = \int_S \vec E_1 \cdot d\vec S = E_1 2 \pi r D = \frac{\lambda_1 D}{\epsilon_0}$
so
$E_1 = \frac{\lambda_1}{2 \pi r \epsilon_0}$
Same thing for the other one.
$E_2 = \frac{\lambda_2}{2 \pi r \epsilon_0}$

At this point I should calculate the velocity starting from the acceleration $a = \frac{e}{m_e}E$, but I'm blocked here. The $E$ in the acceleration is the "total" electric field or it is $E_2$ since the electron is now moving inside the external cylinder?
I know there is this other equation:
$v = \sqrt{\frac{2q(V_1 - V_2)}{m}}$
But in this case I should calculate the two electric potential, which would be:
$\int_A^B \vec E_0 \cdot d\vec l = V(A) - V(B)$
with $A$ being the internal cylinder and $B$ the other one.

 

[mfb]

Working with the potential is the easiest approach here.

What is the field inside a long cylinder with a constant surface charge? (Just the field from this surface charge)

 

[Kernul]

It should be $E = \frac{\sigma R_2}{\epsilon_0 r}$, right?

 

[mfb]

Inside the cylinder?
So at r=0 you have infinite field strength?

 

[Kernul]

I did a drawing of the exercise, just to be sure I'm getting this right.
I used Gauss Theorem to get that. I did like this:
$\Phi_s (E) = \int_S \vec E \cdot d \vec S = E 2 \pi r D = \frac{\sigma 2 \pi R_2 D}{\epsilon_0}$
then
$E 2 \pi r D = \frac{\sigma 2 \pi R_2 D}{\epsilon_0}$
$E = \frac{\sigma 2 \pi R_2 D}{\epsilon_0 2 \pi r D}$
$E = \frac{\sigma R_2}{\epsilon_0 r}$
so I think yes. Unless I got something wrong.

 

[mfb]

For Gauß theorem, you need the charge enclosed by the area you are integrating over. What is the charge in a smaller cylinder inside a charged cylinder?

 

[Kernul]

I'm guessing it should be smaller? Because the charged cylinder act like some kind of insulator to the other one inside?
About the area, I should do something like this then: $E = \int_{R_1}^{R_2} \frac{\sigma R_2}{\epsilon_0 R_1}$
With $R_1$ and $R_2$ being the two points describing the area.

 

[mfb]

Because the charged cylinder act like some kind of insulator to the other one inside?

Forget about the two cylinders for a moment, the problem is earlier.

What is the field inside a single spherical shell with a constant surface charge?

What is the field inside a single cylinder with a constant surface charge?

 

[Kernul]

What is the field inside a single spherical shell with a constant surface charge?

$\vec E = \frac{1}{4 \pi \epsilon_0} \frac{Q}{r^2}$
with $\sigma = \frac{Q}{4 \pi r^2}$ we have then
$\vec E = \frac{\sigma}{\epsilon_0}$

What is the field inside a single cylinder with a constant surface charge?

$\vec E = \frac{1}{4 \pi \epsilon_0} \frac{Q}{r^2}$
with $\sigma = \frac{Q}{2 \pi r h}$ we have then
$\vec E = \frac{\sigma h}{2 r \epsilon_0}$

Is this what you asked for?

 

[mfb]

$\vec E = \frac{1}{4 \pi \epsilon_0} \frac{Q}{r^2}$

Where does that formula come from and what does it calculate?

It does not calculate the electric field inside anything. The right side is also not a vector, so writing E as vector at the left side is not right.

 

[Kernul]

Oh, I'm sorry about the vector thingy.
I know this formula:
$\vec E_0 (\vec r) = \frac {1}{4 \pi \epsilon_0} \int \frac{\sigma(x', y', z')(\vec r - \vec r')}{|\vec r - \vec r'|^3}dS'$ which is just a general way(not only for constant surface charge) and I thought that since $dq = \sigma(x, y, z) dS$ it would end up with the one I wrote ($E_0 = \frac{1}{4 \pi \epsilon_0} \frac{Q}{r^2}$). Is this not correct?

 

[mfb]

It is not correct.
What would r even be there? The position where you want to evaluate the field, or the radius of the sphere?

Anyway, let's stay at the sphere: if it has a radius R, what is the electric field at a radius r where r<R?
You can use Gauß law, but don't blindly plug numbers into a random formula. Think what the formula is doing when, and what the meaning of its components is.

 

[Kernul]

What would r even be there? The position where you want to evaluate the field, or the radius of the sphere?

$r$ is the position where I want to evaluate the field.

Anyway, let's stay at the sphere: if it has a radius R, what is the electric field at a radius r where r<R?

Okay, in $r < R$ we have a $Q = \sigma 4 \pi r^2$
using Gauß law we have
$\Phi = \frac{Q}{\epsilon_0}$
which becomes
$E 4 \pi r^2 = \frac{\sigma 4 \pi r^2}{\epsilon_0}$
$E = \frac{\sigma}{\epsilon_0}$

 

[mfb]

Okay, in $r < R$ we have a $Q = \sigma 4 \pi r^2$

Where would that charge be? All the charge is at the spherical shell, at radius R. There is nothing at smaller radii.

 

[Kernul]

Oh! Because we are talking about surface charge and not volume charge. Because if it was a volume charge then it would have a charge inside the sphere and so it would have $Q = \rho \frac{4}{3} \pi r^3$, right?

 

[mfb]

If it would be a volume charge, yes, but it is not.

 

[Kernul]

So, going back to what you asked me, for $r < R$ the field would be $E = 0$ inside the sphere and $E = \frac{\sigma R^2}{\epsilon_0 r^2}$ on the surface, right?

 

[mfb]

Directly at the surface the field is not well-defined, but outside, yes.
Inside it is zero, right.

So going back to the cylinders: does the charge from the outer cylinder matter for the field inside?

 

[Kernul]

No, it doesn't matter. Only the charge inside matter. This means I only have to calculate the field inside the outer cylinder(which it is the field outside the internal cylinder) $E_1$, but I have the linear charge density $\lambda_1$ and $\lambda_2$, not the surface charge, right?

 

[mfb]

You have the surface charge of the inner cylinder, but a single charged line along its symmetry axis would produce the same field.

 

[Kernul]

So it's okay to write this:
$Q_1 = \lambda_1 D$
$E_1 2 \pi r D = \frac{\lambda_1 D}{\epsilon_0}$
$E_1 = \frac{\lambda_1}{2 \pi r \epsilon_0}$
And I don't have to calculate $E_2$, right?

 

[mfb]

Well, you know $E_2=0$.

Good, now you have everything in place to calculate the velocity.

 

[Kernul]

You said I should use the electric potential, right?
I know that $a = \frac{qE}{m}$, which, in this case, is $a = \frac{e E}{m_e}$
Using the kinetic energy we arrive at $v = \sqrt{\frac{2e(V_1 - V_2)}{m_e}}$.
What I need then is $V_1 - V_2$.
One thing that I know is that $\int_A^B \vec E_0 \cdot d \vec l = V_0(A) - V_0(B)$, with $A$ and $B$ two points. In this case we could say that it's $R_1$ and $R_2$ the two points I want to calculate the electric potentials, right?
So, would it be something like this $\int_{R_1}^{R_2} \vec E_0 \cdot d \vec l = V_0(R_1) - V_0(R_2)$?

 

[mfb]

Right. And you know E as function of radius.

 

[Kernul]

So it would be something like this
$\int_{R_1}^{R_2} \frac{\lambda_1}{2 \pi r \epsilon_0} dl$
$\frac{\lambda_1}{2 \pi \epsilon_0} \int_{R_1}^{R_2} \frac{1}{r} dl$
$\frac{\lambda_1}{2 \pi \epsilon_0} (log|R_2| - log|R_1|)$
Is this correct? To me it's weird that there are logarithms.

 

[mfb]

It is correct, and you get logarithms, right.

 

[Kernul]

Okay, thank you!
But why the problem gave me $\lambda_2$? I didn't use it at all.

 

[mfb]

You also didn't use the length explicitely.
Are there other subproblems? You don't need them here (well, you indirectly used the length in assuming that the cylinder is very long).

 

[Kernul]

No, there were no other subproblems. Maybe the professor put them to trick us.
Anyway, thank you very much! With you I understood a lot of things!