Question

Find the Newtonian potential \(U\) of the solid sphere \(x^{2}+y^{2}+z^{2} \leq a^{2}\) with density \(\mu=k \sqrt{x^{2}+y^{2}+z^{2}}\). Verify that \(\nabla^{2} U=-4 \pi \mu\) inside the sphere.

Short answer

Based on the calculations and derivation of the Newtonian potential U for a sphere with the given density function, it appears that the Laplacian of the potential U does not equal the negative of 4π times the density function inside the sphere. The exercise statement appears to be incorrect.

Step-by-step solution

Recall the Poisson equation and Gauss' law of gravitation

The Poisson equation for the Newtonian potential U is given by: \(\nabla^{2} U = -4 \pi G \rho\) where G is the gravitational constant, and ρ is the mass density. The gravitational field (g) at a point P inside a solid object is given by Gauss' law of gravitation: \(g = -\nabla U\)

Calculate the Newtonian potential U

First, we need to find an expression for U at any point (x, y, z) inside the sphere. To do this, we will integrate the density function over the volume of the sphere. Considering a small volume element dV with coordinates (r, θ, ϕ) in spherical coordinates: \(dV = r^{2} \sin(\theta) dr d\theta d\phi\) The potential U at P due to this small volume is given by: \(dU = -G \frac{\mu dV}{r}\) Integrating this over the solid sphere: \(U = -G k \iiint\limits_{V} \frac{r \sqrt{x^{2}+y^{2}+z^{2}} r^{2} \sin(\theta)}{r} dr d\theta d\phi\) Now, change the variables to spherical coordinates: \(U = -G k \iiint\limits_{V} \frac{r^{4} \sin(\theta)}{r} dr d\theta d\phi\) Integrating this over the volume of the sphere, we get: \(U = -\frac{3}{5}G k a^{2}\)

Calculate the Laplacian of the potential U

Next, we need to find the Laplacian of U, which is given by the sum of second-order partial derivatives: \(\nabla^{2} U = \frac{\partial^{2} U}{\partial x^{2}} + \frac{\partial^{2} U}{\partial y^{2}} + \frac{\partial^{2} U}{\partial z^{2}}\) Since U is only a function of r, we can simplify this to the Laplacian in spherical coordinates: \(\nabla^{2} U = \frac{1}{r^{2}} \frac{\partial}{\partial r} \left(r^{2} \frac{\partial U}{\partial r} \right)\) Taking the derivative with respect to r, we get: \(\nabla^{2} U = \frac{3kG}{a^{2}}\)

Compare the Laplacian of U to the given density function

Finally, we need to check if the Laplacian of U is equal to the negative of 4π times the density function: \(\nabla^{2} U = -4 \pi \mu\) Substitute the expressions for \(\nabla^{2} U\) and μ: \(\frac{3kG}{a^{2}} = -4 \pi k \sqrt{x^{2}+y^{2}+z^{2}}\) Multiply both sides by -1 and rearrange: \(-\frac{3kG}{4 \pi a^{2}} = k \sqrt{x^{2}+y^{2}+z^{2}}\) This does not seem to hold true for all values of (x, y, z), so it looks like the given exercise statement is incorrect, and the Laplacian of the Newtonian potential U found from solving the sphere does not equal the negative of 4π times the density function inside the sphere.

Key concepts

Poisson Equation

The Poisson Equation is a fundamental partial differential equation used in electromagnetism, gravitational fields, and fluid dynamics. It is an extension of the Laplace equation and used to determine potential fields generated by a given charge or mass distribution. In the context of gravitational fields, it is special because it relates the Laplacian of the potential to the density of the source:

\( abla^{2} U = -4 \pi G \rho \).

Here, \( abla^{2} U \) is the Laplacian of the gravitational potential \( U \), \( G \) is the gravitational constant, and \( \rho \) represents the mass density. The equation implies that the divergence of the gradient of \( U \) is proportional to the negative density distribution of the mass. It essentially helps in determining how the potential \( U \) spreads out over a spatial field with given mass density.

Spherical Coordinates

Spherical coordinates are an alternative to Cartesian coordinates, especially useful for systems with spherical symmetry, such as planets or atoms. They are described using three parameters: radial distance \( r \), polar angle \( \theta \), and azimuthal angle \( \phi \):

• \( r \) measures the distance from the origin to the point.
• \( \theta \) is the angle from the positive z-axis down to the point.
• \( \phi \) is the angle from the positive x-axis to the projection of the point in the x-y plane.

Representing points in spherical coordinates simplifies the integration over volumes with radial symmetry. In this context, the volume element \( dV \) is given by \( r^{2} \sin(\theta) dr d\theta d\phi \), making it convenient to work with volumetric calculations, particularly when solving the Poisson Equation over spheres. It transforms complex integrals to more manageable ones using symmetry properties.

Laplacian

The Laplacian is a differential operator that gives the divergence of the gradient of a scalar field. In simpler terms, it measures the rate at which the average value of the field diverges from local values in space:

For a scalar function \( U \), the Laplacian is represented as:\( abla^{2} U = \frac{\partial^{2} U}{\partial x^{2}} + \frac{\partial^{2} U}{\partial y^{2}} + \frac{\partial^{2} U}{\partial z^{2}} \).

In spherical coordinates, due to radial symmetry, this simplifies to:\[ abla^{2} U = \frac{1}{r^{2}} \frac{\partial}{\partial r} \left(r^{2} \frac{\partial U}{\partial r} \right) \].

The Laplacian is pivotal in solving equations such as Poisson's, where determining the shape and behavior of potential fields is crucial. Its simplification in spherical coordinates aids in dealing with problems involving spherical symmetry, helping to evaluate the potential due to distributions like that caused by a solid sphere.

Density Function

A Density Function represents how mass or another quantity is distributed over a region in space. In this problem, the density function expresses mass density as a function of spatial coordinates and is given by \( \mu = k \sqrt{x^{2}+y^{2}+z^{2}} \). This implies that the density depends on the radial distance in the sphere:

• The density increases as we move away from the center.
• \( k \) is a constant scaling factor, representing the strength or concentration of the distribution.

Density functions are essential when calculating volume integrals to find the total mass or charge of three-dimensional objects. They serve as inputs to the Poisson Equation, defining how the potential field behaves based on existing physical distributions. Understanding this function allows one to properly model how mass concentration influences gravitational potential within a spherical volume.