Question regarding Laplace's Equation for regions with charges

In summary: No. The Earnshaw theorem would consider the regions where there are no charges and conclude that there is no stable point for a test charge in those regions. There is not a single second derivative (nor is there a single first derivative). The gradient being zero tells you that all the first derivatives are zero but the Laplace operator acting on the potential being zero tells you that the second derivatives in different directions must have different...
  • #1
Why doesn't the **Laplace's equation**(#\nabla^2V=0#) hold in the region within the sphere when there is a charge inside it ? Is it because #ρ \ne 0# within the sphere and it becomes a **poisson equation**($\nabla^2V=\dfrac{-ρ}{ε_0}$) and changes the characteristics of **Harmonic Solution** (ie),

no maximum or minimum is allowed within the domain of the region in which the laplaces equation is used ? Now this condition becomes redundant because the PDE takes a non zero value .

Can we say that Laplace's equation takes that there is no net charge in the region ,so we add an additional factor of #\dfrac{Q_enc}{4πε_0R²}# to compensate the idea of no charge inside the sphere ?

 
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  • #2
Harikesh_33 said:
Is it because #ρ \ne 0# within the sphere and it becomes a **poisson equation**($\nabla^2V=\dfrac{-ρ}{ε_0}$)
Yes.
 
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  • #3
Orodruin said:
Yes.
Let me get this straight ,so Laplace's equation is used to find a potential function ,now if the potential obeys Laplace's equation then we can find the average potential around a sphere just by taking the potential at the centre of the sphere ,now here comes the catch ,Laplace's equation doesn't conside rthe charge at the centre ,so we add an additional Q_enc /4πε_0R to take the effect of charge at the centre into account .Am i right ?
 
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  • #4
Yes, although it would be more appropriate to say that the Laplace equation follows from the Poisson equation when you set the charge density to zero.
 
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  • #5
Orodruin said:
Yes, although it would be more appropriate to say that the Laplace equation follows from the Poisson equation when you set the charge density to zero.
Yeah but won't the properties of the harmonic function vanish if we use Poisson's equation ?
 
  • #6
Harikesh_33 said:
Yeah but won't the properties of the harmonic function vanish if we use Poisson's equation ?
Orodruin said:
Yes, although it would be more appropriate to say that the Laplace equation follows from the Poisson equation when you set the charge density to zero.
I have a follow up question if you don't mind ,why is earnshaws theroem not true ? I understand that it has something to do with the fact that it's an unstable equilibrium(or it is said so and I don't understand why it is said so ) and potential is maximum for an unstable equilibrium and this isn't
 
  • #7
What do you mean? Earnshaw's theorem says that there is no stable configuration for a set of point charges from electrostatic interactions alone. It is not really what you have been asking about here.

It is also not true that the potential is necessarily at a maximum for an unstable equilibrium. Saddle points are also unstable.
 
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  • #8
Harikesh_33 said:
Yeah but won't the properties of the harmonic function vanish if we use Poisson's equation ?
Yes, the solution is no longer harmonic in general - but it is harmonic in volumes where there is no charge since Poisson's equation then reduces to Laplace's equation.
 
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  • #9
Orodruin said:
What do you mean? Earnshaw's theorem says that there is no stable configuration for a set of point charges from electrostatic interactions alone. It is not really what you have been asking about here.

It is also not true that the potential is necessarily at a maximum for an unstable equilibrium. Saddle points are also unstable.
Sorry for framing the question like that ,to provide some context ,if I am not mistaken ,Laplace's equation is being used to prove Earnshaw's theroem ,But how can we use Laplace's equation here ? Change density is non zero here .
 
  • #10
Earnshaw's theorem is about properties of the electrostatic field in regions with no charge. A practical application is that you can't build a charged-particle trap with electrostatic fields alone. That's why you have, e.g., an additional magnetic field to construct a Penning trap.
 
  • #11
vanhees71 said:
Earnshaw's theorem is about properties of the electrostatic field in regions with no charge. A practical application is that you can't build a charged-particle trap with electrostatic fields alone. That's why you have, e.g., an additional magnetic field to construct a Penning trap.
Aren't we considering the charges placed on the centre and vertices of the cube ? And are you coming to the conclusion that it's a saddle point because both the first and second derivative is zero (follows Laplace's equation )
 
  • #12
Harikesh_33 said:
Aren't we considering the charges placed on the centre and vertices of the cube ? And are you coming to the conclusion that it's a saddle point because both the first and second derivative is zero (follows Laplace's equation )
No. The Earnshaw theorem would consider the regions where there are no charges and conclude that there is no stable point for a test charge in those regions. There is not a single second derivative (nor is there a single first derivative). The gradient being zero tells you that all the first derivatives are zero but the Laplace operator acting on the potential being zero tells you that the second derivatives in different directions must have different signs.
 
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1. What is Laplace's Equation and how does it relate to regions with charges?

Laplace's Equation is a mathematical equation that describes the behavior of electric potential in a region with no charges. However, in regions with charges, the equation is modified to include the presence of electric charges and their effects on the electric potential.

2. What are the boundary conditions for Laplace's Equation in regions with charges?

The boundary conditions for Laplace's Equation in regions with charges are the same as in regions with no charges, which are that the electric potential is continuous and the electric field is perpendicular to the boundaries.

3. How do I solve Laplace's Equation for regions with charges?

The solution to Laplace's Equation for regions with charges can be obtained using various mathematical techniques such as separation of variables, Green's function method, or numerical methods like finite difference or finite element methods.

4. Can Laplace's Equation be used to find the electric field in regions with charges?

Yes, Laplace's Equation can be used to find the electric field in regions with charges by taking the gradient of the electric potential solution obtained from solving the equation.

5. Are there any real-world applications of Laplace's Equation for regions with charges?

Yes, Laplace's Equation for regions with charges has many real-world applications, such as in electrostatics, electromagnetism, and fluid dynamics. It is used to model the behavior of electric fields and potentials in various systems, such as electronic circuits, conductors, and semiconductor devices.

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