Question

Question regarding cylinders and radial electric fields

Answer 1

You understand that the recent increase in U.S. natural gas prices is expected to shunt demand for gas as utilities use more coal to generate power. Such utilities are interested in the use of electrostatic precipitators, which use electric forces to remove pollutant particles from smoke, in particular in the smokestacks of coal-burning power plants. One form of precipitator you are studying consists of a vertical, hollow, metal cylinder with a thin wire, insulated from the cylinder, running along its axis. A large potential difference is established between the wire and the outer cylinder, with the wire at lower potential. This sets up a strong radial electric field directed inward. The field produces a region of ionized air near the wire. Smoke enters the precipitator at the bottom, ash and dust in it pick up electrons, and the charged pollutants are accelerated toward the outer cylinder wall by the electric field. Suppose the radius of the central wire is 9.0 x 10–5 m, the radius of the cylinder is 0.14 m, and the potential difference of 5 x 104 V is established between the wire and the cylinder. Also assume that the wire and cylinder are both very long in comparison to the cylinder radius. You are asked:

(a) What is the magnitude of the electric field midway between the wire and the cylinder wall?

(b) What magnitude of charge must a 3.00 x 10–8 kg ash-particle have if the electric field computed in part (a) is to exert a force ten times the weight of the particle?

@Mashy

Answer 2

@Mashy Now, there are two approaches that I have found in my research. One claims that the field is uniform and radial, therefore the problem can be solved using
\[V_2-V_1=-\int\limits E dr\]

However, another claim is that the field is not uniform. How should I approach this?

Answer 3

your questions are very nicely worded :P .. What grade questions are these?

Clearly radial cylindrical fields are non uniform.

Answer 4

They are university-engineering level :)

And that would make sense because at the longer ends, the field should be weaker.

Answer 5

Okay..

you should know the expression for the cylindrical field

\[E = \frac{\lambda}{2 \pi \epsilon_0 R}\]

Answer 6

Okay and so
\[\lambda=\frac{Q}{L}\] right? But is it the length of the whole cylinder?

Answer 7

Since it is infinitely long (very long)
its better to keep the answer in lamda itself.

Now our job is to calculate lambda..

We have the knowledge of potential difference.

so you can do the integral.. plug in the values of V2, V1, and calculate Lambda

Answer 8

How would I go about setting up this integral?
\[\lambda=\int\limits \frac{ E }{ 2\pi \epsilon_0R }dR\]?

Answer 9

you already did.. now integrate it from R1 (the radius of the wire) to R2 (radius of the pipe)

Answer 10

\[\lambda=\frac{ E }{ 2\pi \epsilon_0 }\ln(\frac{ R_2 }{ R_1 }) \]

Answer 11

Wait.. I overlooked.. why is there an E? :P

It should be lamda

Answer 12

I solved for lambda in the first equation you gave me and then integrated with respect to R >_< lol

Answer 13

Wait, I'm not actually sure what I did.. lol

Answer 14

I think I accidentally swapped the variables. It should be
E = lambda and the rest

Answer 15

yes.. just swap E and Lamda and you should be fine!

Answer 16

Sweet. And for the second part it's jsut F=qE? F=mg, so mg=qE and just solve for q?

Answer 17

Well 10mg since it says the F is 10 times the weight

Answer 18

10 times the weight

Answer 19

yea yea.. correct

Answer 20

you are all set on reducing pollution.. (Well theoretically :P)

Answer 21

Thank you! Do you have time for one more problem? I think it should be simple but I'm stuck (probably because it's in the wee hours of the evening :P)

Answer 22

ok shoot.. lets do one more before I go for lunch

Answer 23

@Mashy Actually, I just realized that my problem contains actual constants, so I think I'm suppose to have a numerical answer rather than an expression. I don't know lambda. They've also given me the potential difference..