A fully ionized gas containing a single species of ion with...

A fully ionized gas containing a single species of ion with charge $Z|e|$ and atomic weight $A$ is in equilibrium in a uniform gravitational field $g$. The gas is isothermal with temperature $T$ and there is thermal equilibrium between the ions and the electrons. The gas has a low enough density that local interactions between the particles can be neglected. (a) Show that to avoid charge separation there must be a uniform electric field $E$ given by $$ E=-\frac{\left(A m_{\mathrm{p}}-m_e\right)}{(1+Z)|e|} g, $$ where $m_{\mathrm{p}}$ and $m_e$ are the proton and electron masses respectively. (b) Show that the above equation is also valid if the plasma is not isothermal. (Hint: Treat each component $i$ as an ideal gas subject to the equation of hydrostatic equilibrium $$ \frac{d p_i}{d x}=n_i F_{i x} $$ where $p_i$ is the partial pressure of the $i$ th component, $n_i$ is its number density, and $F_{i x}$ is the total force per particle in the $x$ direction.) (c) The equation in (a) is also valid throughout the sun where $\mathbf{E}$ and g are now directed radially. Show that the charge on the sun is given approximately by $$ Q=\frac{A}{1+Z} \frac{G M m_p}{|e|}, $$ where $M$ is the mass of the sun. (d) For the sun $M=2 \times 10^{33}$ grams. If the composition of the sun were pure hydrogen, what would be $Q$ in coulombs? Given this value of $Q$, is the approximation that there is no charge separation a good one?

A fully ionized gas containing a single species of ion with charge $Z|e|$ and atomic weight $A$ is in equilibrium in a uniform gravitational field $g$. The gas is isothermal with temperature $T$ and there is thermal equilibrium between the ions and the electrons. The gas has a low enough density that local interactions between the particles can be neglected.
(a) Show that to avoid charge separation there must be a uniform electric field $E$ given by
$$
E=-\frac{\left(A m_{\mathrm{p}}-m_e\right)}{(1+Z)|e|} g,
$$
where $m_{\mathrm{p}}$ and $m_e$ are the proton and electron masses respectively.
(b) Show that the above equation is also valid if the plasma is not isothermal. (Hint: Treat each component $i$ as an ideal gas subject to the equation of hydrostatic equilibrium
$$
\frac{d p_i}{d x}=n_i F_{i x}
$$
where $p_i$ is the partial pressure of the $i$ th component, $n_i$ is its number density, and $F_{i x}$ is the total force per particle in the $x$ direction.)
(c) The equation in (a) is also valid throughout the sun where $\mathbf{E}$ and g are now directed radially. Show that the charge on the sun is given approximately by
$$
Q=\frac{A}{1+Z} \frac{G M m_p}{|e|},
$$
where $M$ is the mass of the sun.
(d) For the sun $M=2 \times 10^{33}$ grams. If the composition of the sun were pure hydrogen, what would be $Q$ in coulombs? Given this value of $Q$, is the approximation that there is no charge separation a good one?

Key Concepts

Electric Field in Gravitational Equilibrium

An electric field can arise in a plasma when gravitational forces act differently on ions and electrons, due to their mass disparity. This electric field, often termed the ambipolar electric field, develops to counteract charge separation by creating an additional force on the charged particles. Its role is crucial to maintain overall quasi-neutrality of the plasma while balancing the differing gravitational pulls on ions and electrons.

Ideal Gas Behavior

The ideal gas approximation assumes that the particles within the plasma interact weakly enough that they can be treated as non-interacting point particles, obeying the ideal gas law. This assumption permits the use of simple relations between pressure, number density, and temperature for both ions and electrons, which are foundational when deriving equilibrium conditions in plasmas.

Plasma Quasi-neutrality and Charge Separation

Quasi-neutrality is the principle that in a plasma, the densities of positive and negative charges are nearly equal, so that any local charge separation is extremely small. This near-neutrality is maintained through the rapid redistribution of charges via the electric field. Understanding this concept is key to analyzing how even weak electric fields, established to prevent charge separation, can significantly influence the pressure and density profiles in an astrophysical plasma.

Hydrostatic Equilibrium

Hydrostatic equilibrium refers to the balance between the inward force of gravity and the outward pressure gradient in a fluid. In the context of a plasma composed of ions and electrons, each species independently satisfies hydrostatic equilibrium under the combined influence of gravitational and other forces, such as electric forces. This principle is critical for understanding how pressure gradients form and maintain stability in stars and other astrophysical objects.

Thermal Equilibrium

Thermal equilibrium implies that all components of a system, such as ions and electrons in a plasma, share the same temperature. This common temperature ensures that the pressure contributions from the different species can be described using the same thermal energy scale, which simplifies the analysis of forces such as those due to gravity and electric fields within the plasma.

Transcript

00:01 Hello, the question is taken from chapter 16 of your physics book.

00:08 So given that if in suppose both earth and moon carried the same net charge.

00:16 So given that the charge on if this is earth and this is moon, both carry the same charge.

00:28 So let me take that charge is equal to capital q and capital q.

00:34 And but after that spread uniformly over their surfaces because if we consider them like conductor on conductor the whole of charges is distributed over its surface how large q need to be so that systems electric and gravitational potential energies had equal magnitude and under these conditions so let me evaluate the first part so we need to evaluate the value of q so we need to evaluate the value of at which the electrical potential energy and gravitational potential energy are equal.

01:12 So let me take the distances between them is r.

01:16 And mars of earth is capital m, mars of moon is small m.

01:21 The earth, moon.

01:26 Hope that you understand everything.

01:28 Let me erase it because we know that charge on both of them are equal.

01:33 So let me write here also.

01:38 Charge on earth and moon are equal okay let me write the given that condition so f e is equal electrical force and gravitational force and gravitational forces between two charges are you since the charges are same so q square by four pi epsilon naught not not force but potential energy sorry so that is u e u g so that is 4 pi epsilon out r and gravitational potential energy is g capital m is the mass of earth small m is the mass of moon divided by r so this r will be cancelled out so in the end we get the value of charge is equal to square root g capital m small m into 4 pi epsilon o so let me check what is the mass of earth i think that is mass of earth is 6 10 to the power 24 kg so capital m is 6 into 10 to the power 24 kg and mass of moon let me write it as mass of moon is 7 .35 into 10 to the power 22 so that is 7 .35 into 10 to the power 22 kg that is the mass of moon and what other thing we want nothing so that q will be equal to square root and the value of gravitational constant is equal to 6 .67 into 10 to the power minus 11 newton meter square per kilogram square and the value of 4 pi epsilon note is 1 by 9 into 10 to the power 9 so let me substitute all those values so that are 6 .67 into 10 to the power minus 11 mass of earth is 10 to the power 24 into mass of moon is 7 .35 into 10 to the power 22 divided by 9 9 into 10 to the power 9 and that one is also in square root so let me evaluate this value so let me take my calculator in order to evaluate that value so the values are 6 .67 e minus 11 multiply 6 e 24 multiply 7 .35 e 22 divided by 99 to the power 0 .5 .5 i think that charge is too much large...

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