Potential and E field for a non homogeneous charge Density

In summary, the conversation discusses the solution to Laplace's equation with a specific set of boundary conditions. The solution involves two functions, one for the interior and one for the exterior, with constants c and d. The setup is confirmed to be correct and it is also mentioned that another solution with constants k and l is also possible.
  • #1
Diracobama2181
75
2
Homework Statement
Suppose you have an infinite slab of thickness 2a has a volume charge density ρ(x) given by $$ρ(x) = ρ_0sin(πx/a) , |x| ≤ a$$, and
$$ρ(x) =0 ,|x| > a$$
where ρ0 and a are positive constants. The geometry of this
system is such that x = 0 is the central plane contained inside the slab with
the x−axis being perpendicular to it. In addition, take this plane to be the
potential reference plane; i.e., φ(x = 0) = 0. For the above charge distribution,
calculate the potential φ(x) and the electric field E(x) everywhere in space.
Relevant Equations
$$\nabla^2V=-\frac{\rho}{\epsilon}$$
Based on the conditions, I found that $$V(x)=\frac{a^2}{\pi^2} ρ_0sin(πx/a)$$ would be a solution to Laplace's equation for $$|x|\leq a$$
and $$V(x)=cx+d$$, where c and d are constants. From the boundary conditions, $$\frac{dV(a)}{dx}=\frac{a}{\pi} ρ_0cos(πa/a)=ac$$, $$c=\frac{a\rho}{\pi}$$ and $$V(a)=ca+d=\frac{a^2}{\pi^2} ρ_0sin(πa/a)$$, $$V=\frac{a\rho}{\pi}x+\frac{a^2\rho}{\pi}$$ for $$x>a$$ (a similar setup would find V(x) for x<-a). Does my setup seem correct? Also, wouldn't $$V(x)=\frac{a^2}{\pi^2} ρ_0sin(πx/a)+kx+l$$ also be a solution for constants k and l? Thank you.
 
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  • #2
You can't avoid a piecewise definition here. One function for the interior, one function for the outside, that is fine.
 

1. What is the definition of potential in relation to non-homogeneous charge density?

Potential is a measure of the work required to move a unit charge from one point to another in an electric field. In the case of non-homogeneous charge density, the potential varies at different points due to the varying distribution of charges.

2. How is the electric field calculated for a non-homogeneous charge density?

The electric field at a point is given by the gradient of the potential at that point. In the case of non-homogeneous charge density, the potential is a function of position and thus the electric field is also a function of position.

3. Can the potential and electric field be determined analytically for a non-homogeneous charge density?

In most cases, it is not possible to determine the potential and electric field analytically for a non-homogeneous charge density. Instead, numerical methods such as finite difference or finite element methods are used to approximate the solutions.

4. How does the presence of non-homogeneous charge density affect the behavior of the electric field?

Non-homogeneous charge density can cause the electric field to vary in magnitude and direction at different points. This can result in complex patterns and behavior of the electric field, making it difficult to predict and analyze.

5. Is it possible to have a uniform electric field in the presence of non-homogeneous charge density?

No, it is not possible to have a uniform electric field in the presence of non-homogeneous charge density. The electric field will vary in magnitude and direction at different points due to the uneven distribution of charges.

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