Poissons equation for the yukawa potential is given by;
Phi(r)=-(G*M exp(-alpha*r))/r
what is the analogue of poissons equation for this potential.
To find the analog of the Poisson's equation for the Yukawa potential, we will start by determining the Laplacian of the potential function ϕ(r).
The Laplacian (∇^2) of a scalar function ϕ(r) in three dimensions is given by:
∇^2ϕ = (1/r^2) (∂/∂r) (r^2 (∂ϕ/∂r)) + (1/r^2 sinθ) (∂/∂θ) (sinθ (∂ϕ/∂θ)) + (1/r^2 sin^2θ) (∂^2ϕ/∂φ^2),
where r, θ, and φ are the spherical coordinates.
For the Yukawa potential ϕ(r) = -(G*M*exp(-α*r))/r, we need to calculate its Laplacian by taking the derivatives with respect to the spherical coordinates.
First, we differentiate ϕ with respect to r:
(∂ϕ/∂r) = -(G*M*exp(-α*r))/r^2 + α*(G*M*exp(-α*r))/r
Next, we differentiate r^2 (∂ϕ/∂r) with respect to r:
(∂/∂r) (r^2 (∂ϕ/∂r)) = 2*r (∂ϕ/∂r) + r^2 (∂^2ϕ/∂r^2)
Now, we differentiate ϕ with respect to θ and φ:
(∂/∂θ) (sinθ (∂ϕ/∂θ)) = sinθ (∂^2ϕ/∂θ^2) + cosθ (∂ϕ/∂θ)
(∂^2ϕ/∂φ^2) = 0 since it does not depend on φ
Substituting these derivatives into the Laplacian expression, we have:
∇^2ϕ = (1/r^2) [2*r ((-G*M*exp(-α*r))/r^2 + α*(G*M*exp(-α*r))/r) + r^2 (∂^2ϕ/∂r^2)] + (1/r^2 sinθ) [sinθ (∂^2ϕ/∂θ^2) + cosθ (∂ϕ/∂θ)] + 0
Simplifying the expression further, we get:
∇^2ϕ = -2*alpha*(G*M*exp(-alpha*r))/r - (-alpha^2)*(G*M*exp(-alpha*r))/r + 2/r^2 * (-G*M*exp(-alpha*r))/r^2 - (G*M*exp(-alpha*r))/r^3*sinθ * cosθ
Finally, rearranging the terms, we obtain the analog of Poisson's equation for the Yukawa potential:
∇^2ϕ = (G*M*exp(-alpha*r))(alpha^2/r - 2(alpha/r^2 + 1/r^3) - sinθ*cosθ/r^3)
This equation describes the behavior of the Yukawa potential in terms of its Laplacian (∇^2ϕ) in spherical coordinates.