Electric potential, solution to Laplace's Eq.

In summary, the potential outside a radially symmetric charge distribution of total charge q, given by V = q/(4*pi*epsilon_0*r), is a solution to Laplace's Equation. Instead of solving Laplace's Equation directly, it is easier to demonstrate that this solution satisfies the equation. This can be done by taking the 2nd derivative of V and showing that it equals 0. In spherical coordinates, the Laplacian of a scalar function f is equal to \frac{\partial^2 f}{\partial r^2}.
  • #1
nsatya
26
0

Homework Statement



Prove that the potential outside of any radially symmetric charge distribution of total charge q, given by,

V = q/(4*pi*epsilon_0*r)

is a solution to Laplace's Equation.

Hint: Only a masochist would solve this problem by solving Laplace's Equation. It is much easier to demonstrate that this solution is a solution to Laplace's equation.



Homework Equations





The Attempt at a Solution



I first tried to plug V into Laplace's Equation del^2 V = 0. Since V only depends on r in this case, I thought I could just take the 2nd derivative of V and show that it is 0. I got to this point but could not go much further with it. Any help would be appreciated.
 
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  • #2
Hint: what is the Laplacian of any scalar function [tex]f[/tex] in spherical coordinates? Is the radial component really just [tex]\frac{\partial^2 f}{\partial r^2}[/tex] ? :wink:
 

FAQ: Electric potential, solution to Laplace's Eq.

What is electric potential?

Electric potential is the amount of electric potential energy per unit charge at a given point in an electric field. It represents the work required to move a unit charge from one point to another in an electric field.

How is electric potential related to Laplace's equation?

Laplace's equation is a mathematical equation that describes the electric potential in a region with no charge. It states that the Laplacian of the electric potential is equal to zero, indicating that the potential does not change in the absence of charge.

What is the solution to Laplace's equation?

The solution to Laplace's equation for a given boundary condition is known as the electric potential function. This function describes the distribution of electric potential in a region with no charge.

How is the electric potential function determined?

The electric potential function is determined by solving Laplace's equation using techniques such as separation of variables, Green's functions, or numerical methods. The solution is then evaluated at the boundary conditions to obtain the specific electric potential function for that system.

How is electric potential different from electric field?

Electric potential is a scalar quantity that describes the energy per unit charge at a point, while electric field is a vector quantity that describes the force per unit charge at a point. Electric potential is related to electric field through the gradient operator, where electric field is the negative gradient of the electric potential.

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