Electromagnetism Q: Charge Density & Total Charge Calculation

[Monster007]

Hi, studying for exam and checking if i am doing this question correctly ,

Q: Cylinder radius R=1m length L=2m exists in free space with no other charges nearby.
Cylinder volume is charged with uniform charge Q= 7C.

(a)Find the charge density in Coulombs per cubic metre.
(b) charge distribution now changes to p = a.s (s=radial distance from central axis of cylinder)
find the total charge.


Answer: (a) I think p=Q/V as uniform charge so;

p=7/(volume cylinder) = 7/2(pi)

(b) total charge =Q = integral(charge dens . volume)

hence Q= ∫∫∫(a.r).(2∏r²).r.dr.dθ.dz (where 0<θ<2∏, 0< z < 2, 0<r<1)

and doing the math i get : Q = (8∏².a)/5


There are no answers to the questions in this course for some reason so hard to know if I am right or on completely wrong track so any help would be appreciated cheers

 

[Philip Wood]

Your integrand, I think, is wrong. You are, I think, intending to integrate ρ dV, in which dV is the volume element. But instead of using the volume element, which is rdθ dr dz, you are using 2πr² times this. You'd have spotted this if you'd considered the UNITS of your integrand.

The extra 2πr² probably arises because you're mixing integration going right back to the basic volume element with integration by cylindrical shell - perfectly o.k in itself, but you can't mix the two methods - you're integrating too many times!

 

[andrien]

volume element is simply 2πrdrdz,what you have really written there(I integrated over angle because of symmetry)