Potential in the three regions of an infinite slab

  • #1
workhorse123
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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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  • #2
Welcome to PF!
If possible, please type in your work using Latex, rather than posting snapshots of your work. Additional guidelines for posting in the homework forums can be found here.

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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1. What is the potential in the middle region of an infinite slab?

The potential in the middle region of an infinite slab is constant and equal to the potential at the surface of the slab.

2. How does the potential vary in the region outside the slab?

Outside the slab, the potential decreases linearly with distance from the slab. The potential at infinity is zero.

3. What is the potential inside the slab?

Inside the slab, the potential is constant and equal to the potential at the surface of the slab.

4. How does the potential change at the boundaries of the slab?

At the boundaries of the slab, there is a discontinuity in the potential. The potential jumps by an amount proportional to the surface charge density at the boundary.

5. How does the potential change with the thickness of the slab?

The potential inside the slab remains constant regardless of the thickness of the slab. However, the magnitude of the potential jump at the boundaries of the slab is directly proportional to the surface charge density and inversely proportional to the thickness of the slab.

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