CP A Parallel Universe. Imagine a parallel universe in which...

CP A Parallel Universe. Imagine a parallel universe in which the electric force has the same properties as in our universe but there is no gravity. In this parallel universe, the sun carries charge $Q,$ the earth carries charge $-Q,$ and the electric attraction between them keeps the earth in orbit. The earth in the parallel universe has the same mass, the same orbital radius, and the same orbital period as in our universe. Calculate the value of $Q$ . (Consult Appendix F as needed.)

Key Concepts

Coulomb's Law

Coulomb's Law describes the force between two point charges. It states that the magnitude of the electric force between charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them, with the constant of proportionality involving the permittivity of free space. This law is essential for understanding how electric forces can mimic gravitational forces in settings such as the one described.

Centripetal Force and Circular Motion

In orbital mechanics, an object moving in a circular orbit experiences a centripetal force that keeps it in its path. This force is proportional to the mass of the orbiting object and the square of its velocity, and inversely proportional to the radius of the orbit. Equating this centripetal force to the electric force (as given by Coulomb's Law) allows one to analyze and constrain the conditions for stable orbit in contexts where gravity is replaced by another force, such as electricity.

Electrostatic Force as a Central Force

The concept of a central force is one that is directed along the line connecting two interacting bodies. In a scenario where gravitation is absent, if charges are arranged to provide an attractive force, the resulting electric force acts as a central force. This concept is crucial for substituting gravitational interaction with an electric force in orbital dynamics, thereby enabling analyses of orbits based on electric interactions.

Permittivity of Free Space

Permittivity of free space is a fundamental physical constant that appears in Coulomb's Law. It quantifies the ability of free space to permit electric field lines, thereby influencing the magnitude of the electric force between charges. An understanding of this constant is critical when performing calculations involving electric forces over large distances, such as determining the required charge for maintaining orbital motion.

Transcript

00:01 In this exercise, we are working in a scenario on which there is no gravitational force, and the earth has a total charge of minus q, and the sun has a total charge of plus q, and the earth moves around the sun in a uniform circular movement, and the only force between these two bodies is the electric force.

00:24 This exercise asks us to evaluate what should be the charge q, of the earth and the sun so that the earth has the same translational period as we have today.

00:44 Okay, so here i wrote some important quantities we must know.

00:48 So here in blue i wrote the mass of the earth.

00:52 I wrote the radius of the earth's orbit around the sun and i wrote c as a translational period.

01:04 Okay, so since we are working with a uniform circular movement, we know that the force, the radial force on the uniform circular movement is equal to m, only to m is the mass of the object that is moving around the circle, which in our case is the mass of the earth, times the tangential velocity of the orbit squared over the radius of the orbit.

01:45 This is equal to the radio force.

01:49 Since we don't have any gravitational force acting here and the only force that is in the radial direction is the electric force, so this should be equal to the charge of the earth kill times the electric field, generated by the sun.

02:11 Okay, so we know that the electric field, we can calculate the electric field of the sun, in this case by using kulom's law and treated sun as a point charge, since the radius of the earth is much bigger than the radius of the sun, so the sun in this scenario can be treated as a point charge, and we have that the force that the sun, the sun, acts on the earth is equal to q squared times k on each k is the k -constant over the radius of the orbit squared, and we saw that this should be equal to the mass of the earth times the orbits velocity squared over r...

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