Understanding Self Energy

Self energy is a concept in physics that refers to the energy required to assemble a system from its constituent parts. This energy is stored within the system due to the interactions between its parts. In the context of gravitation, self energy is the work done in assembling a mass distribution by bringing infinitesimal mass elements from infinity to their final positions within the system.

Self Energy in Gravitational Systems

For a gravitational system, the self energy is associated with the gravitational potential energy of the system. It is the work done against the gravitational force to assemble the mass distribution.

Formula for Self Energy

The self energy ($U$) of a spherical mass distribution of radius $R$ and total mass $M$ can be calculated using the formula:

$$ U = -\frac{3}{5}\frac{G M^2}{R} $$

where $G$ is the gravitational constant.

Calculation of Self Energy

To calculate the self energy of a system, one must integrate the work done to bring each infinitesimal mass element $dm$ from infinity to its final position within the mass distribution. The work done $dW$ to bring a mass element $dm$ to a distance $r$ from the center of the distribution is given by:

$$ dW = -\frac{G M(r) dm}{r} $$

where $M(r)$ is the mass enclosed within radius $r$ at the moment the mass element $dm$ is brought in.

The total self energy is then the integral of $dW$ over the entire mass distribution:

$$ U = \int dW = -\int_0^R \frac{G M(r) dm}{r} $$

Examples

Example 1: Self Energy of a Uniform Sphere

For a uniform sphere of mass $M$ and radius $R$, the self energy can be calculated by considering the mass distribution to be spherically symmetric. The mass enclosed within a radius $r$ is:

$$ M(r) = \frac{4}{3}\pi r^3 \frac{M}{\frac{4}{3}\pi R^3} = \frac{M r^3}{R^3} $$

Substituting $M(r)$ into the integral for self energy, we get:

$$ U = -\int_0^R \frac{G \left(\frac{M r^3}{R^3}\right) dm}{r} $$

Since $dm = \rho dV = \rho 4\pi r^2 dr$ and $\rho = \frac{M}{\frac{4}{3}\pi R^3}$, we can integrate to find the self energy:

$$ U = -\int_0^R \frac{G M r^2}{R^3} 4\pi r^2 dr = -\frac{3}{5}\frac{G M^2}{R} $$

Example 2: Self Energy of a Thin Shell

For a thin spherical shell of mass $M$ and radius $R$, the mass distribution is concentrated at radius $R$. The self energy is calculated by considering that all mass elements are brought to the same radius:

$$ U = -\frac{G M^2}{2R} $$

Differences and Important Points

Aspect	Uniform Sphere	Thin Shell
Mass Distribution	Uniform throughout the volume	Concentrated at the surface
Self Energy Formula	$U = -\frac{3}{5}\frac{G M^2}{R}$	$U = -\frac{G M^2}{2R}$
Work Done per Mass Element	Depends on the radius $r$ at which $dm$ is located	Same for all mass elements since $r = R$
Integration Limits	From $0$ to $R$	Not applicable (mass is at $R$)
Physical Interpretation	Work done to assemble a solid sphere	Work done to assemble a spherical shell

In summary, self energy is a measure of the energy stored in a system due to its internal gravitational interactions. It is an important concept in astrophysics and cosmology, where it plays a role in the dynamics and evolution of celestial bodies and structures. Understanding self energy is crucial for students preparing for exams in physics, particularly in the field of gravitation.