Question

(a) Show that the gravitational potential due to a uniform disc of radius \(a\) and mass \(M\), centred at the origin, is given for \(ra\) by $$ \frac{G M}{r}\left[1-\frac{1}{4}\left(\frac{a}{r}\right)^{2} P_{2}(\cos \theta)+\frac{1}{8}\left(\frac{a}{r}\right)^{4} P_{4}(\cos \theta)-\cdots\right] $$ where the polar axis is normal to the plane of the disc. (b) Reconcile the presence of a term \(P_{1}(\cos \theta)\), which is odd under \(\theta \rightarrow \pi-\theta\), with the symmetry with respect to the plane of the disc of the physical system. (c) Deduce that the gravitational field near an infinite sheet of matter of constant density \(\rho\) per unit area is \(2 \pi G \rho\).

Short answer

The potential for \( r < a \) and \( r > a \) is given using Legendre polynomials expansion. The symmetry is maintained despite the presence of \( P_1(\cos \theta) \). The gravitational field near an infinite sheet is \( 2 \pi G \rho \).

Step-by-step solution

Understanding the Gravitational Potential Expression

The gravitational potential \( \Phi \) due to a mass distribution can generally be expressed using spherical harmonics and Legendre polynomials. For a uniform disc, this involves integrating over the disc's surface.

Gravitational Potential Inside the Disc (r < a)

For \( r < a \), the potential is given by \( \frac{2 G M}{a}\left[1-\frac{r}{a}P_{1}(\cos \theta)+\frac{1}{2}\left(\frac{r}{a}\right)^{2}P_{2}(\cos \theta)-\frac{1}{8}\left(\frac{r}{a}\right)^{4}P_{4}(\cos \theta)+\cdots\right]. \)To derive this, use the fact that inside the disc, the potential must account for every point within the radius \( a \), contributing to the total potential.

Gravitational Potential Outside the Disc (r > a)

For \( r > a \), the potential is given by \( \frac{G M}{r}\left[1-\frac{1}{4}\left(\frac{a}{r}\right)^{2}P_{2}(\cos \theta)+\frac{1}{8}\left(\frac{a}{r}\right)^{4}P_{4}(\cos \theta)-\cdots\right]. \)This is derived by expanding the gravitational potential in Legendre polynomials and using the boundary conditions.

Addressing Symmetry and the P1 Term

The term \( P_{1}(\cos \theta) \) (which is \( \cos \theta \)) appears to break the symmetry because it is odd under \( \theta\rightarrow\pi-\theta \). However, in the context of the problem, the coefficient for \( P_1 \) (i.e., \( \frac{r}{a} \)) can account for the disc’s mass distribution, maintaining the overall physical symmetry.

Gravitational Field Near an Infinite Sheet

Using the symmetry and distribution properties, deduce that the field near an infinite sheet of matter with constant density \( \rho \) per unit area results in a gravitational field strength of \( 2 \pi \mathrm{G} \rho \). This can be derived from Gauss’s law for gravity.

Key concepts

Legendre polynomials

Legendre polynomials, denoted as \(P_n(\cos\theta)\), are a series of orthogonal polynomials that arise in solving various problems in physics and engineering, particularly those involving spherical coordinates. They are commonly used in gravitational potential expansions due to their useful properties. These polynomials help simplify complex expressions by breaking down the potential into components based on angular dependence.
One of their key characteristics is orthogonality, which means that integrating the product of two different Legendre polynomials over the range \( [ -1, 1 ] \) results in zero. This property is crucial for simplifying integrals in potential theory.
For the gravitational potential of a disc, the use of Legendre polynomials allows us to express the potential as a sum of terms, each term corresponding to a different polynomial with coefficients that depend on the ratios \( \frac{r}{a} \) and \( \frac{a}{r} \). This approach leverages the orthogonality and recurrence relationships of Legendre polynomials to handle the angular part of the problem efficiently.

spherical harmonics

Spherical harmonics, often denoted as \( Y_{lm}(\theta, \phi) \), are functions that arise in the solution to Laplace's equation in spherical coordinates. They are used in various fields including gravitational potential theory, quantum mechanics, and electromagnetism.
They provide a convenient basis for expanding any function defined on the surface of a sphere, simplifying the representation of the harmonic functions. In gravitational potential problems, spherical harmonics help to express the potential in terms of angular coordinates (\(\theta\) and \(\phi\)) and radial distance.
In the case of a uniform disc's gravitational potential, spherical harmonics help separate the radial and angular dependencies. Since the disc has rotational symmetry around a single axis, only certain terms (e.g., \( Y_{1m} \) ) contribute, simplifying the series expansion. This approach handles the angular dependence through spherical harmonics while focusing on radial terms for the full potential solution.

Gauss's law for gravity

Gauss's law for gravity is a fundamental law in gravitational theory that relates the gravitational flux through a closed surface to the mass enclosed by that surface. It is mathematically expressed as: \[ \oint_{} \mathbf{g} \cdot d\mathbf{A} = -4 \pi G M_{\text{enc}} \] where \( \mathbf{g} \) is the gravitational field, \( d\mathbf{A} \) is a vector representing an infinitesimal area on the surface, and \( M_{\text{enc}} \) is the enclosed mass.
Gauss's law is particularly useful in problems with high symmetry, such as spherical or planar mass distributions. It simplifies the computation of gravitational fields because you only need to consider the total enclosed mass instead of performing complex integrals.
In the context of an infinite sheet of matter, Gauss's law helps deduce the gravitational field near the sheet. By considering a Gaussian surface (cylindrical) that straddles the sheet, we determine that the gravitational flux is proportional to the mass per unit area \( \rho \). This leads to the result that the gravitational field strength near an infinite sheet with constant density is \( 2 \pi G \rho \).

symmetry in physical systems

Symmetry plays a crucial role in simplifying physical problems. Symmetry can refer to geometric, rotational, or even mirror symmetry. When a system is symmetric, certain terms in mathematical expansions might cancel out or simplify significantly.
For the gravitational potential of a disc, the symmetry around the disc's central plane is essential. This symmetry implies that the physical properties of the system, including the gravitational potential, should remain unchanged under transformations like \( \theta \rightarrow \pi - \theta \). This is why an odd term like \( P_1(\cos \theta) \) must be carefully examined. Despite appearing to break symmetry individually, the overall physics respects the system's symmetry.
Symmetry considerations help in reconciling the presence of apparently asymmetric terms. They ensure that the final potential or field respects the geometry and mass distribution of the physical system. This approach simplifies understanding and solving many gravitational field problems.