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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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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FAQ: Potential in the three regions of an infinite slab

What is an infinite slab in the context of electrostatics?

An infinite slab in electrostatics refers to a theoretical construct where a material with a certain charge distribution extends infinitely in two dimensions (x and y), but has a finite thickness in the third dimension (z). This simplification allows for easier mathematical analysis of the electric potential and field.

How is the electric potential inside the infinite slab determined?

The electric potential inside an infinite slab is determined by solving Poisson's equation, which relates the electric potential to the charge density within the slab. The solution typically involves integrating the charge density over the volume of the slab and applying boundary conditions to find the potential at any point inside.

What boundary conditions are used for solving the potential in an infinite slab?

For an infinite slab, the boundary conditions often assume that the potential approaches a constant value far away from the slab. Additionally, symmetry considerations are used, such as the potential being an even function of the distance from the center of the slab if the charge distribution is uniform.

How does the potential vary outside the infinite slab?

Outside the infinite slab, the potential typically decreases with distance from the slab. The exact form of this decrease depends on the charge distribution within the slab. For a uniformly charged slab, the potential outside can be found by integrating the contributions from all the infinitesimal charge elements within the slab.

What role does the thickness of the slab play in determining the potential?

The thickness of the slab is crucial in determining the potential distribution. Inside the slab, the potential depends on the distance from the center and the charge density. The thicker the slab, the more complex the potential distribution can be, as it involves integrating over a larger volume. Outside the slab, the thickness influences how quickly the potential falls off with distance.

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