Potential in the three regions of an infinite slab

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The discussion focuses on the boundary conditions for electric fields and potentials in the context of an infinite slab. It emphasizes that while electric potential can have an arbitrary reference point, the electric field cannot be freely chosen and must adhere to specific conditions. Participants explore justifying the condition that the electric field at z = 0 is zero due to the symmetry of the charge distribution. The conversation also encourages the use of LaTeX for clarity in mathematical expressions. Overall, the thread aims to clarify the application of boundary conditions in electrostatics.
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Homework Statement
The charge density in the region
-z’<z<z’
depends only on z; that is,
p=p’cos(pi z/z’)
where p’ and z’ are constants. Determine the potential in all regions of space
Relevant Equations
Poisons equation, laplace equation
for the boundary conditions for this problem I understand that Electric field and Electric potential will be continuous on the boundaries.
I also know that I can set the reference point for Electric potential, wherever it is convenient. This gives me the fifth boundary condition. I am confused at where I find the last boundary condition.

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You cannot arbitrarily choose a "zero point" for the electric field like you can with the potential. However, can you justify the condition ##E_{\text{at} \, z = 0} = 0## from the symmetrical nature of the charge distribution in this problem?
 
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I want to find the solution to the integral ##\theta = \int_0^{\theta}\frac{du}{\sqrt{(c-u^2 +2u^3)}}## I can see that ##\frac{d^2u}{d\theta^2} = A +Bu+Cu^2## is a Weierstrass elliptic function, which can be generated from ##\Large(\normalsize\frac{du}{d\theta}\Large)\normalsize^2 = c-u^2 +2u^3## (A = 0, B=-1, C=3) So does this make my integral an elliptic integral? I haven't been able to find a table of integrals anywhere which contains an integral of this form so I'm a bit stuck. TerryW

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