Use Gauss' Law to calculate the electrostatic potential for this cylinder

In summary, the conversation was about solving the Laplacian equation and finding the solution for V(r, phi). The solution was found to be V(r, phi) = a + b.lnr + a series of terms involving r, phi, and constants. The speaker then asked for help with using boundary conditions to find the constants. The other person suggested using a solution of the form V(r, phi) = V_0f(r)sin(phi) with a boundary condition of V(R, phi) = V_0sin(phi). The speaker also asked if the same boundary conditions applied for both inner and outer potential and if phi could be equal to phi at r=R. The response was that since the cylinder is non-conduct
  • #1
Reg_S
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Homework Statement
An infinitely long hollow (non conducting) circular cylinder of radius R fixed at potential V =V•sin(phi) .
Relevant Equations
Using cylinder coordinates with z axis as a symmetric axis , argue V is independent of Z and V(r, -phi)= -V(r, phi)
b) Find electrostatic potential inside and outside of the cylinder
I solved laplacian equation. and got the solution of V(r, phi) = a. +b.lnr + (summation) an r^n sin(n phi +alpha n ) + (summation) bn r ^-n sin( n phi +beta n)
 
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  • #2
Please help me how to use BCs and find the constant,
 
  • #3
The boundary condition is [itex]V(R, \phi) = V_0\sin \phi[/itex] (please don't use the same symbol for an unknown function and a given constant value). That immediately suggests trying a solution of the form [tex]V(r, \phi) = V_0f(r)\sin \phi[/tex] with [itex]f(R) = 1[/itex].
 
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  • #4
Thank you, Is it same BCs for inner and outer potential? just using relative term? Can we do (Phi)in = (phi)out at r=R?
 
  • #5
The cylinder is non-conducting, so the potential is continuous across it.
 
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FAQ: Use Gauss' Law to calculate the electrostatic potential for this cylinder

1. What is Gauss' Law?

Gauss' Law is a fundamental law in electromagnetism that relates the electric flux through a closed surface to the charge enclosed by that surface. It is often used to calculate the electric field or potential at a point due to a given distribution of charges.

2. How do you use Gauss' Law to calculate the electrostatic potential?

To use Gauss' Law to calculate the electrostatic potential, you first need to determine the charge distribution and choose a closed surface that encloses the charge. Then, you can use the formula V = Q/(4πε₀r) to calculate the potential at a given distance from the charge, where Q is the total charge enclosed by the surface, ε₀ is the permittivity of free space, and r is the distance from the charge to the point where the potential is being calculated.

3. What is the formula for calculating the electrostatic potential using Gauss' Law?

The formula for calculating the electrostatic potential using Gauss' Law is V = Q/(4πε₀r), where V is the potential at a point, Q is the total charge enclosed by a closed surface, ε₀ is the permittivity of free space, and r is the distance from the charge to the point where the potential is being calculated.

4. Can Gauss' Law be used for any charge distribution?

Yes, Gauss' Law can be used for any charge distribution as long as the charge is enclosed by the chosen closed surface. However, for more complex charge distributions, it may be necessary to use a combination of Gauss' Law and other techniques to calculate the potential.

5. What is the significance of using Gauss' Law to calculate the electrostatic potential?

Using Gauss' Law to calculate the electrostatic potential allows us to simplify complex problems and make use of symmetry in charge distributions. It also helps us to understand the relationship between the electric field and potential, and how they are affected by the distribution of charges.

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