Surface density of the charges induced on the bases of the cylinder

AI Thread Summary
The discussion revolves around the calculation of surface charge density induced on the bases of a cylinder, with the correct formula identified as σ = ε₀E(ε - 1)/ε. Initial attempts to derive this involved confusion over the electric fields E_in and E_out, particularly the relationship E_in = E_out/ε. The use of Gauss' law clarified that the electric field produced by a surface charge σ is E = σ/(2ε₀). The final derivation correctly incorporates the dielectric constant ε, confirming it is dimensionless, leading to the conclusion that the solution is valid. The importance of accurately applying Gauss' law and the definitions of electric fields in dielectric materials is emphasized.
rokiboxofficial Ref
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Homework Statement
The dielectric cylinder is in an external uniform electric field E, which is parallel to the axis of the cylinder. Find the surface density of the charges induced on the bases of the cylinder. Dielectric constant of the cylinder material is equal to ε. The height of a cylinder is much less than the radius of its bases.
Relevant Equations
##E_{in} = E_{out} - E_{ind}##
##E_{in} = \frac{E_{out}}{\varepsilon}##
The correct answer to this problem is: ##\sigma = \varepsilon_0E\frac{\varepsilon-1}{\varepsilon}##
Here is my attempt to solve it, please tell me what is my mistake?
##E_{in} = E_{out} - E_{ind}##
##E_{ind} = E_{out} - E_{in}##
##E_{in} = \frac{E_{out}}{\varepsilon}##
##E_{ind} = E_{out} - \frac{E_{out}}{\varepsilon}##
##E = \frac{\sigma}{2\varepsilon_0\varepsilon}##
##E_{ind} = E_{ind+} + E_{ind-}##
##E_{ind} = \frac{\sigma}{2\varepsilon_0\varepsilon} + \frac{\sigma}{2\varepsilon_0\varepsilon} = \frac{\sigma}{\varepsilon_0\varepsilon}##
##\sigma = E_{ind}\varepsilon_0\varepsilon##
##\sigma = \left( E_{out} - \frac{E_{out}}{\varepsilon} \right)\varepsilon_0\varepsilon##
##\sigma = E_{out}\varepsilon_0\varepsilon- \frac{E_{out}\varepsilon_0\varepsilon}{\varepsilon}##
##\sigma = E_{out}\varepsilon_0\varepsilon- E_{out}\varepsilon_0##
##\sigma = E_{out}\varepsilon_0\left( \varepsilon- 1 \right)##
 
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I think ##~E_{in} = \frac{E_{out}}{\varepsilon}~## should be ##~E_{in} = \frac{\varepsilon_0 E_{out}}{\varepsilon}~## otherwise the dimensions are incorrect. See if that fixes the problem.
 
kuruman said:
I think ##~E_{in} = \frac{E_{out}}{\varepsilon}~## should be ##~E_{in} = \frac{\varepsilon_0 E_{out}}{\varepsilon}~## otherwise the dimensions are incorrect. See if that fixes the problem.
No, it doesn't fix it:

##E_{in} = \frac{\varepsilon_0E_{out}}{\varepsilon}##
##E_{ind} = E_{out} - E_{in}##
##E_{ind} = E_{out} - \frac{\varepsilon_0E_{out}}{\varepsilon}##
##\sigma = \left( E_{out} - \frac{\varepsilon_0E_{out}}{\varepsilon} \right)\varepsilon_0\varepsilon##
##\sigma = E_{out}\varepsilon_0\varepsilon- \frac{\varepsilon_0E_{out}\varepsilon_0\varepsilon}{\varepsilon}##
##\sigma = E_{out}\varepsilon_0\varepsilon- E_{out}{\varepsilon_0}^2##
##\sigma = E_{out}\varepsilon_0\left( \varepsilon- \varepsilon_0 \right)##

And the formula ##E_{in} = \frac{E_{out}}{\varepsilon}## was given to me in the training material for the task, I think that it is most likely correct.
 
rokiboxofficial Ref said:
The correct answer to this problem is: ##\sigma = \varepsilon_0E\frac{\varepsilon-1}{\varepsilon}##
Here is my attempt to solve it, please tell me what is my mistake?
##E_{in} = E_{out} - E_{ind}##
##E_{ind} = E_{out} - E_{in}##
##E_{in} = \frac{E_{out}}{\varepsilon}##
##E_{ind} = E_{out} - \frac{E_{out}}{\varepsilon}##
##E = \frac{\sigma}{2\varepsilon_0\varepsilon}##
##E_{ind} = E_{ind+} + E_{ind-}##
##E_{ind} = \frac{\sigma}{2\varepsilon_0\varepsilon} + \frac{\sigma}{2\varepsilon_0\varepsilon} = \frac{\sigma}{\varepsilon_0\varepsilon}##
##\sigma = E_{ind}\varepsilon_0\varepsilon##
##\sigma = \left( E_{out} - \frac{E_{out}}{\varepsilon} \right)\varepsilon_0\varepsilon##
##\sigma = E_{out}\varepsilon_0\varepsilon- \frac{E_{out}\varepsilon_0\varepsilon}{\varepsilon}##
##\sigma = E_{out}\varepsilon_0\varepsilon- E_{out}\varepsilon_0##
##\sigma = E_{out}\varepsilon_0\left( \varepsilon- 1 \right)##
I believe the mistake occurs where you wrote $$E = \frac{\sigma}{2\varepsilon_0\varepsilon}$$ for the electric field produced by ##\sigma## on one of the surfaces. You can treat the surface as an infinite plane. The electric field produced by an infinite plane of surface charge ##\sigma## can be found using Gauss' law.
 
kuruman said:
I think ##~E_{in} = \frac{E_{out}}{\varepsilon}~## should be ##~E_{in} = \frac{\varepsilon_0 E_{out}}{\varepsilon}~## otherwise the dimensions are incorrect. See if that fixes the problem.
Looks like ##\varepsilon## is being used for the relative permittivity, so is dimensionless.
 
TSny said:
I believe the mistake occurs where you wrote $$E = \frac{\sigma}{2\varepsilon_0\varepsilon}$$ for the electric field produced by ##\sigma## on one of the surfaces. You can treat the surface as an infinite plane. The electric field produced by an infinite plane of surface charge ##\sigma## can be found using Gauss' law.
Yes, you are right! Thank you a lot!
Is this solution a correct?
By Gauss' law:
##\frac{q}{\varepsilon_0} = ES_{full}##
##\frac{q}{\varepsilon_0} = 2ES##
##E = \frac{q}{2\varepsilon_0S}##
##E = \frac{\sigma}{2\varepsilon_0}##
##E_{ind} = E_{ind+} + E_{ind-}##
##E_{ind} = \frac{\sigma}{2\varepsilon_0} + \frac{\sigma}{2\varepsilon_0} = \frac{\sigma}
{\varepsilon_0}##
##\sigma = E_{ind}\varepsilon_0##
##E_{ind} = E_{out} - \frac{E_{out}}{\varepsilon}##
##\sigma = \left( E_{out} - \frac{E_{out}}{\varepsilon} \right)\varepsilon_0##
##\sigma = E_{out}\varepsilon_0- \frac{E_{out}\varepsilon_0}{\varepsilon}##
##\sigma = E_{out}\varepsilon_0\left( 1 - \frac{1}{\varepsilon}\right)##
##\sigma = E_{out}\varepsilon_0\left( \frac{\varepsilon}{\varepsilon} - \frac{1}{\varepsilon}\right)##
##\sigma = E_{out}\varepsilon_0\left(\frac{\varepsilon - 1}{\varepsilon}\right)##
 
rokiboxofficial Ref said:
Yes, you are right! Thank you a lot!
Is this solution a correct?
By Gauss' law:
##\frac{q}{\varepsilon_0} = ES_{full}##
##\frac{q}{\varepsilon_0} = 2ES##
##E = \frac{q}{2\varepsilon_0S}##
##E = \frac{\sigma}{2\varepsilon_0}##
##E_{ind} = E_{ind+} + E_{ind-}##
##E_{ind} = \frac{\sigma}{2\varepsilon_0} + \frac{\sigma}{2\varepsilon_0} = \frac{\sigma}
{\varepsilon_0}##
##\sigma = E_{ind}\varepsilon_0##
##E_{ind} = E_{out} - \frac{E_{out}}{\varepsilon}##
##\sigma = \left( E_{out} - \frac{E_{out}}{\varepsilon} \right)\varepsilon_0##
##\sigma = E_{out}\varepsilon_0- \frac{E_{out}\varepsilon_0}{\varepsilon}##
##\sigma = E_{out}\varepsilon_0\left( 1 - \frac{1}{\varepsilon}\right)##
##\sigma = E_{out}\varepsilon_0\left( \frac{\varepsilon}{\varepsilon} - \frac{1}{\varepsilon}\right)##
##\sigma = E_{out}\varepsilon_0\left(\frac{\varepsilon - 1}{\varepsilon}\right)##
Looks good.
 
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haruspex said:
Looks like ##\varepsilon## is being used for the relative permittivity, so is dimensionless.
Yes. ##\varepsilon## is given to be the "dielectric constant", which is the same as the relative permittivity.
 
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