Is the Potential V=x^2-y^2+z^2 Valid in Free Space?

In summary, the conversation discusses the validity of the potential distribution equation V= x2-Y2+Z2 in free space and how to calculate the charge density from it. It is determined that the given potential does not satisfy Laplace's equation, but may still follow Poisson's equation. However, due to dimensional inaccuracy, any answer derived from the equation cannot be correct.
  • #1
knavneet
3
0
Can anyone tell me if the potential distribution equations V= x2-Y2+Z2
holds in free space, if not, please help me to find the charge density from the given potential .
 
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  • #2
Hint:What does Poisson's Equation tell you?
 
  • #3
Del2V=2 for the given V, so the given potential doesn't satisfy Laplace equation, So i think this potential is not possible in free space. But how to calculate charge density then?
 
  • #4
Yes, Laplace's equation is not valid. However Poisson's equation is. As gabbagabbahey said, What does Poisson's equation tell you?
 
  • #5
According to Poisson equation Del2V=-charge density / epslion ,
But in our problem Del2V=2 , So charge density = 2*epsilon , Is it the answer now ?
 
  • #6
On the basis of what you have said, no. The equation, V= x2-y2+z2 is dimensionally incorrect. Therefore any answer derived from it cannot be correct.
 

FAQ: Is the Potential V=x^2-y^2+z^2 Valid in Free Space?

What is charge density?

Charge density is a measure of the amount of electric charge per unit volume in a given material or space. It is typically represented by the Greek letter rho (ρ) and is expressed in units of coulombs per cubic meter (C/m³).

How is charge density calculated?

Charge density is calculated by dividing the total amount of charge in a given volume by the volume itself. For example, if a cube with sides of 1 meter contains 2 coulombs of charge, the charge density would be 2 C/m³.

What is the relationship between charge density and electric field?

Charge density and electric field are directly related. The electric field, represented by the symbol E, is equal to the charge density multiplied by a constant known as the permittivity of free space (represented by the Greek letter epsilon, ε). This relationship is expressed as E = ρ/ε.

How does charge density affect the behavior of electric fields?

Charge density plays a key role in determining the strength and direction of electric fields. In general, the higher the charge density in a given space, the stronger the electric field will be. Additionally, the distribution of charge within a space can also affect the behavior of electric fields.

Can charge density change over time?

Yes, charge density can change over time depending on the movement or flow of electric charges within a given space. For example, if more charges are added to a specific volume, the charge density will increase. Similarly, if charges are removed from a space, the charge density will decrease.

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