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The gravitational force between two point masses separated by a distance \(r\) is proportional to \(1 / r^{2},\) just like the electric force between two point charges. Because of this similarity between gravitational and electric interactions, there is also a Gauss's law for gravitation. (a) Let \(\vec{g}\) be the acceleration due to gravity caused by a point mass \(m\) at the origin, so that \(\vec{g}=-\left(G m / r^{2}\right) \hat{r}\) . Consider a spherical Gaussian surface with radius \(r\) centered on this point mass, and show that the flux of \(\vec{g}\) through this surface is given by $$\oint \vec{g} \cdot d \vec{A}=-4 \pi G m$$ (b) By following the same logical steps used in Section 22.3 to obtain Gauss's law for the electric field, show that the flux of \(\vec{g}\) through any closed surface is given by $$\oint \vec{g} \cdot d \vec{A}=-4 \pi G M_{\mathrm{encl}}$$ where \(M_{\text { encl }}\) is the total mass enclosed within the closed surface.

Step by step solution

01

Understand the gravitational field

The gravitational field \( \vec{g} \) due to a point mass \( m \) at the origin is given by \(-\vec{g} = \frac{Gm}{r^2} \hat{r}\), where \( G \) is the gravitational constant, \( r \) is the radial distance from the mass, and \( \hat{r} \) is the radial unit vector.

02

Define the Gaussian surface and flux

Consider a spherical Gaussian surface of radius \( r \) centered on the point mass. The gravitational flux through this surface is calculated using \(-\oint \vec{g} \cdot d\vec{A}\), where \( d\vec{A} \) is an infinitesimal area vector on the spherical surface directed radially outward.

03

Evaluate the surface integral for flux

Because \( \vec{g} = -\frac{Gm}{r^2} \hat{r} \) and \( d\vec{A} = dA \hat{r} \), the dot product \( \vec{g} \cdot d\vec{A} = \left(-\frac{Gm}{r^2} \hat{r}\right)\cdot\left(dA \hat{r}\right) = -\frac{Gm}{r^2} dA \). Thus, \(-\oint \vec{g} \cdot d\vec{A} = -\frac{Gm}{r^2} \oint dA \).

04

Solve the integral over the sphere

The integral \( \oint dA \) over the sphere with area \( A = 4\pi r^2 \) becomes \( \oint dA = 4\pi r^2 \). Substituting this into the flux expression gives \( -\frac{Gm}{r^2} \times 4\pi r^2 = -4\pi Gm \).

05

Recognize generalization to Gauss's law for gravitation

Using the similar logic as for electric fields, if any closed surface encloses a total mass \( M_{\text{encl}} \), we can generalize that the gravitational flux through this surface is \(-\oint \vec{g} \cdot d\vec{A} = -4\pi GM_{\text{encl}} \), resulting from the superposition principle of gravitation over enclosed mass.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gravitational Field

Understanding the gravitational field is crucial as it describes how a mass affects the space around it. The gravitational field, represented by \( \vec{g} \), due to a point mass \( m \) at the origin, is given by \(-\vec{g} = \frac{Gm}{r^2} \hat{r} \). This formula tells us that the gravitational field decreases with the square of the distance \( r \) from the mass. Here, \( G \) is the universal gravitational constant, and \( \hat{r} \) is a unit vector pointing radially away from the mass.

The negative sign indicates that the gravitational field direction is inward, towards the mass responsible for creating it.

In essence, it gives us the acceleration due to gravity experienced by any object placed close to this mass, and it behaves similarly to the electric field created by point charges.

Let's explore other elements that amplify our comprehension of gravitational interactions.

Gaussian Surface

A Gaussian surface is an imaginary closed surface used in analyzing fields like gravity or electromagnetism. It helps to apply Gauss's law effectively. For gravitational fields, consider a spherical Gaussian surface with radius \( r \) centered around a point mass. This allows for simplification in calculations by symmetry.

The gravitational field at every point on this sphere is identical in magnitude and direction determined by \( \vec{g} = -\frac{Gm}{r^2} \hat{r} \).

By employing a Gaussian surface, we can easily calculate the gravitational flux through any symmetrical configuration, streamlining the application of Gauss's law for gravitation.

Surface Integral

The concept of a surface integral plays a pivotal role in computing the interaction through surfaces in space. When calculating gravitational flux, the surface integral of the gravitational field \( \vec{g} \) over a closed surface \( S \) results in \( \oint \vec{g} \cdot d\vec{A} \).

Here, \( d\vec{A} \) represents an infinitesimal area vector on the surface that points outward. Specifically, for a spherical surface, this results in simple calculations due to symmetry: \( \vec{g} \cdot d\vec{A} = \left(-\frac{Gm}{r^2} \hat{r}\right) \cdot \left(dA \hat{r}\right) = -\frac{Gm}{r^2} dA \).

Evaluating this integral over the entire surface provides a measure of the gravitational influence contained within that closed area. Surface integral thus serves as a bridge between local field properties and their overall manifestations.

Flux

Flux quantifies the total effect of a field passing through a surface. In gravitation, it specifically measures the number of gravitational field lines permeating a surface. For a spherical surface centered on a point mass, we use the expression: \(-\oint \vec{g} \cdot d\vec{A} = -\frac{Gm}{r^2} \times 4\pi r^2 = -4\pi Gm \).

The negative sign shows that this inward flux correlates with the inward force direction of gravitational fields.

Simply put, gravitational flux is a way of accounting for the total gravitational effect exerted by a mass on a surrounding surface, highlighting the integral aspect of field dynamics.

Superposition Principle

The superposition principle is indispensable when dealing with multiple masses influencing a gravitational field. It states that the total gravitational field due to several masses is the vector sum of fields due to each individual mass.

In line with Gauss's law for gravitation, if a closed surface encloses multiple masses, the gravitational flux through this surface is \(-\oint \vec{g} \cdot d\vec{A} = -4\pi G M_{\text{encl}} \), where \( M_{\text{encl}} \) is the sum of masses inside.

This principle ensures that gravitational effects can be easily calculated even in complex systems, by considering each mass's effect on the total field independently.