Line charge within a charged shell (basket case)

• hashbrowns808
In summary: The fields will average to the charge density of the insulating material. I think your best bet would be to look for a more in depth tutorial on the subject.
hashbrowns808
1. Homework Statement
An infinite line of charge with linear density λ1 = 6.8 μC/m is positioned along the axis of a thick insulating shell of inner radius a = 2.5 cm and outer radius b = 4.8 cm. The insulating shell is uniformly charged with a volume density of ρ = -667 μC/m3.

1) What is λ2, the linear charge density of the insulating shell?

-3.518247622μC/m

2) What is Ex(P), the value of the x-component of the electric field at point P, located a distance 8.1 cm along the y-axis from the line of charge?

0 N/C

3) What is Ey(P), the value of the y-component of the electric field at point P, located a distance 8.1 cm along the y-axis from the line of charge?

This is where I'm stuck.

Homework Equations

Gauss' law ∫E⋅dA=Q_enc/∈
I know there will be integrations in there but due to a week of snow days the instructor hasn't covered any of that, so I'm not sure if there is something else I'm missing, and I'm not finding any good material that I can extract something from.

The Attempt at a Solution

The first section to the homework was similar, but with a conducting shell. I figured that out with the help of Michael Van Biezen videos, but I don't get how to use charge densities, or how to integrate what.

E∫(πR2)L = ∫dQ/∈ (I'm using ∈ for epsilon naught)
dQ=λdV
dV=∫(from a-b)πL(Ra2-Rb2)
dQ=λπL∫(Ra2-Rb2)
EπR2 = (λπL∫(Ra[/SUB2]-Rb2))/∈
E = (λL∫ab(Ra2-Rb2))/R∈

That's what I have written on my paper. I'm just totally lost and desperate at this point. Like I said my teacher is expecting way too much without much instruction, and I'm sorry for short comings, but I just really need help.

Thanks!

Attachments

• h4_cylinder.png
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Last edited:
Ey(P) = 2*k(this is 9.0E9)*(Charge density of outer shell * area of outer cylinder + lambda of line) all divided by radius (.081m) let me know if this works out for you

marchcha said:
Ey(P) = 2*k(this is 9.0E9)*(Charge density of outer shell * area of cylinder (use length 1) + lambda of line) all divided by radius (.081m) let me know if this works out for you
******length of one is not even needed disregard that

marchcha said:
******length of one is not even needed disregard that

I got:

Ey(P) =( 2(8.99810^9)(-667π(.0482-.0252))/.081
= -780964101685
and it was incorrect.

hashbrowns808 said:
I got:

Ey(P) =( 2(8.99810^9)(-667π(.0482-.0252))/.081
= -780964101685
and it was incorrect.
what you are trying to do in this problem is act like the system is a point charge so take the lambda you found in the beginning and add it to the lambda given multiply it by two and k and then divide by the radius (0.081)

marchcha said:
what you are trying to do in this problem is act like the system is a point charge so take the lambda you found in the beginning and add it to the lambda given multiply it by two and k and then divide by the radius (0.081)

Ok so (((6.8x10^-6)+(-3.518x10^-6))2(8,99x10^9))/.081
=728522.9629

Got it. Thank you so much! I think I'll have to work out that integral backwards.
I'm asking this before I've figured out how to arrive at the formula you gave me, but why can I treat it as two point charges? The x components of the shell cancel, and y components < 0 (along y axis), is it just that the y-comp fields end up averaging to λ1? Does anyone know of a good link where I can learn more about this?

I use Flipit physics, and I find it difficult to follow at best, and like I said we're cramming 3 sections into 1 week, where we'd normally be doing half that, and the teacher hasn't been able to give us much help.

Again thanks Marchcha for the help.

hashbrowns808 said:
Ok so (((6.8x10^-6)+(-3.518x10^-6))2(8,99x10^9))/.081
=728522.9629

Got it. Thank you so much! I think I'll have to work out that integral backwards.
I'm asking this before I've figured out how to arrive at the formula you gave me, but why can I treat it as two point charges? The x components of the shell cancel, and y components < 0 (along y axis), is it just that the y-comp fields end up averaging to λ1? Does anyone know of a good link where I can learn more about this?

I use Flipit physics, and I find it difficult to follow at best, and like I said we're cramming 3 sections into 1 week, where we'd normally be doing half that, and the teacher hasn't been able to give us much help.

Again thanks Marchcha for the help.
Hey as a user of flipit physics I feel your pain. You treat it as two point charges because it is outside of the conductor.

1. What is a line charge within a charged shell?

A line charge within a charged shell, also known as a "basket case," refers to a situation where a long, thin wire with a uniform charge density is placed inside a conducting sphere that has a different, but uniform, charge density. This creates a system with two different types of charge distributions.

2. How is the electric field calculated for a line charge within a charged shell?

The electric field for a line charge within a charged shell can be calculated by using the superposition principle, which states that the electric field at a point is the sum of the individual electric fields produced by each charge distribution separately. The electric field for the wire can be calculated using the equation for the electric field of a line charge, while the electric field for the shell can be calculated using the equation for the electric field of a spherical shell.

3. What is the significance of a line charge within a charged shell in physics?

The concept of a line charge within a charged shell is important in understanding how different charge distributions interact and affect the overall electric field. This concept is also used in various applications such as in the design of capacitors and in solving problems related to electrostatics.

4. How does the electric potential vary in a line charge within a charged shell?

The electric potential for a line charge within a charged shell varies depending on the distance from the wire and the distance from the center of the shell. At points outside the shell, the potential is constant and equal to the potential of the wire. At points inside the shell, the potential varies with distance from the center of the shell according to the electric potential of a spherical shell.

5. Are there any real-life examples of a line charge within a charged shell?

Yes, there are several real-life examples of a line charge within a charged shell. One common example is a coaxial cable, where a thin wire with a uniform charge is placed inside a cylindrical conductor with a different, but uniform, charge. Another example is a parallel plate capacitor, where the two plates act as the charged shells and the space between them contains a line charge in the form of opposite charges on the plates.

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