Балкански 2010, Теория — Задача 1. Gravitational Field

Concours Général de Physique “Minko Balkanski” · 8 май 2010 г. · 8 т.

Условие

In this part the Sun and the Earth are supposed to be homogeneous spheres. Using an electrostatic analogy, we are going to find the gravitational field, the gravitational potential and the gravitational energies.

1. Find the electrical field $\vec{E}(A)$, in a point $A$, created by a point charge $q$ fixed at point $O$, using: $OA = r$ and $\vec{u}_r = \frac{\overrightarrow{OA}}{r}$.

2. We consider a spherical surface $S$, centered in $O$.

2. (a) Cite Gauss’ theorem for the electrical field.

2. (b) Determine the flow $\Phi$ of the vector $\vec{E}$ going out of $S$. Note $Q_{int}$ the total charge inside the spherical surface $S$.

3. We have an uniformly charged ball of radius $R$, with a total charge $Q$.

3. (a) Determine the electrical field $\vec{E}(r)$ created by this ball at a distance $r$ from its center $O$ for $r$ between $0$ and $\infty$, as a function of $Q$, $\vec{u}_r$, $r$ and $R$. Consider two cases $0 < r < R$ and $R < r < +\infty$.

4. (a) Provide the formula of the relationship between the electrostatic energy interaction of the ball in question $(3.a)$ and a point charge $q$ put at a distance $r$ from its center $O$, and the electrostatic potential $U(r)$ of the ball, found in the previous question.

4. (b) Find this energy $E_1$ (supposed zero in infinity).

5. (a) Remind the expression of the volume density of the electrostatic energy in a point $M$ where the field is $\vec{E}(M)$.

5. (b) Show that the electrostatic energy of the ball can be written as $E_2 = \frac{kQ^2}{4\pi\epsilon_0 R}$.

6. Now you can use the formal analogy between the electrical and gravitational field:

6. (a) Give the expression of the gravitational field $\vec{g}(A)$, in a point $A$ created by a point mass $m$ fixed in $O$, using: $OA = r$ and $\vec{u}_r = \frac{\overrightarrow{OA}}{r}$.

6. (b) Write explicitly the analogous couples (example: charge and mass).

6. (c) Consider a homogeneous ball with radius $R$ and mass $M$. Determine the gravitational field $\vec{g}(r)$ that this ball creates at a distance $r$ from its center $O$, for $r$ between $0$ and $+\infty$

6. (d) Numerical application: calculate the intensity $g_{SE}$ of the gravitational field created by the Sun on Earth’s surface and also the intensity $g_{ES}$ of the gravitational field created by the Earth on the Sun’s surface.

6. (e) Determine the gravitational energy of interaction between the Earth, assumed as a point, and the Sun.

6. (f) Numerical application.

7. Find the proper energy of the Sun and the Earth noted respectively, $E_{2S}$ and $E_{2T}$.

8. Numerical application: calculate these energies and find the energy $E_{2TS}$ of the system Earth-Sun neglecting (in the interaction energy) their radius compared to the distance between the planets. Comment your result.

Решение

Оригинал в Архива: CGP_2010_en.pdf