Relating Volume Charge Density and Linear Charge Density

In summary: Thanks for the help!In summary, the linear charge density of the insulating shell is given by λ = 2π(b^2 - a^2)ρ, where ρ is the volume density of the shell and b and a are the outer and inner radii, respectively. Removing the 2 in the formula gives the correct answer.
  • #1
acdurbin953
42
0

Homework Statement


An infinite line of charge with linear density λ1 = 6.7 μC/m is positioned along the axis of a thick insulating shell of inner radius a = 2.4 cm and outer radius b = 4.7 cm. The insulating shell is uniformly charged with a volume density of ρ = -722 μC/m3.

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

Homework Equations


(Vouter shell-Vinner shell)*ρ <-- current one I'm using, but I could be wrong.

The Attempt at a Solution


My approach has been to find a bridge between the two densities by relating the volume to the length, but in every attempt I've made I've run into the problem of what to do about the length variable, as the shell is infinite. This is a gauss's law problem, so I've also tried using that to get rid of length variable - however I realized I didn't have the E-field.

With the current equation I'm using, I've again run into not knowing what to do about the length of the cylindrical shell.

ETA - Okay, I've now tried this:

ρ=Q/(2pi(b2-a2)L) and Q=ρ*V=λ*L

So, λ*L=(2pi(b2-a2)L)*ρ, and the lengths will cancel

λ=(2pi(b2-a2))*ρ

Answer is wrong though.
 
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  • #2
acdurbin953 said:
λ=(2pi(b2-a2))*ρ
Looks right to me. What do you get numerically? Do you know what the answer is supposed to be?
 
  • #3
acdurbin953 said:

Homework Statement


An infinite line of charge with linear density λ1 = 6.7 μC/m is positioned along the axis of a thick insulating shell of inner radius a = 2.4 cm and outer radius b = 4.7 cm. The insulating shell is uniformly charged with a volume density of ρ = -722 μC/m3.

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

Homework Equations


(Vouter shell-Vinner shell)*ρ <-- current one I'm using, but I could be wrong.

The Attempt at a Solution


My approach has been to find a bridge between the two densities by relating the volume to the length, but in every attempt I've made I've run into the problem of what to do about the length variable, as the shell is infinite. This is a gauss's law problem, so I've also tried using that to get rid of length variable - however I realized I didn't have the E-field.

With the current equation I'm using, I've again run into not knowing what to do about the length of the cylindrical shell.

ETA - Okay, I've now tried this:

ρ=Q/(2pi(b2-a2)L) and Q=ρ*V=λ*L

So, λ*L=(2pi(b2-a2)L)*ρ, and the lengths will cancel

λ=(2pi(b2-a2))*ρ

Answer is wrong though.

There is no "2" in the Volume of a cylinder ... it is Area x Length.
 
  • #4
lightgrav said:
There is no "2" in the Volume of a cylinder ... it is Area x Length.
Ah yes - I didn't notice the 2.
 
  • #5
haruspex said:
Looks right to me. What do you get numerically? Do you know what the answer is supposed to be?

I did (-722E-6)(2pi(0.0472-0.0242)) = -7.41 μC, I input -7.41 as the units were already included and the answer was wrong. I have no access to the answer.
 
  • #6
Ah! I have no idea how I didn't notice that. Removed the 2 and the answer was correct. Thank you!

I must've carried the 2 around after playing with using the surface area to solve.
 

1. What is volume charge density?

Volume charge density, denoted by ρ, is a measure of the electric charge per unit volume of a material. It is expressed in units of coulombs per cubic meter (C/m^3).

2. How is volume charge density related to linear charge density?

Volume charge density is equal to the linear charge density (λ) multiplied by the cross-sectional area (A) of the material. In other words, ρ = λA.

3. What is linear charge density?

Linear charge density, denoted by λ, is a measure of the electric charge per unit length of a one-dimensional object, such as a wire. It is expressed in units of coulombs per meter (C/m).

4. How is linear charge density related to the total charge of an object?

The total charge Q of an object is equal to the linear charge density (λ) multiplied by the length (L) of the object. In other words, Q = λL.

5. Can the volume charge density of a material vary?

Yes, the volume charge density of a material can vary depending on the amount and distribution of electric charge within the material. It can also be influenced by external factors such as electric fields or the presence of other materials.

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