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Coulomb Field 1

Under the influence of the coulomb field of charge + Q charge -q is moving around it in an elliptical Orbit

The Force:

  • The positive charge (+Q) creates a Coulomb field, which is a force field that exerts a repulsive force on the negative charge (-q).
  • This force is described by Coulomb’s Law: F = k * (q1 * q2) / r^2, where:
    • F is the force between the charges
    • k is Coulomb’s constant (a physical constant)
    • q1 and q2 are the magnitudes of the charges (in this case, +Q and -q)
    • r is the distance between the centers of the charges

The Orbit:

  • Due to the attractive force between opposite charges, -q is not simply repelled away from +Q.
  • Instead, the balance between this force and the inertia of -q causes it to move in a closed path, which in this case, is an ellipse.
  • One focus of the ellipse coincides with the location of the positive charge (+Q).

Important Consequences:

  • Constant Angular Momentum: The total angular momentum of the system (charge -q) about +Q remains constant throughout the motion. This is because the net torque acting on -q due to the central force from +Q is zero.
  • Varying Speed: The distance between -q and +Q changes as it moves along the ellipse. This leads to a variation in the linear speed of -q. It’s fastest when closest to +Q (at the perigee) and slowest when farthest (at the apogee).
  • Non-Constant Linear Momentum: As the speed of -q changes, its linear momentum also changes throughout the orbit.

Additional Notes:

  • This scenario is similar to the motion of planets around the Sun, where the Sun acts as the central mass and the planets experience a gravitational force instead of an electrostatic force.
  • Knowing the specific properties of the system (masses, charges, initial conditions), detailed calculations can be made to predict the motion of -q along the ellipse using the laws of motion and Kepler’s laws of planetary motion.

Why is the torque 0 ?

A central force, like the Coulomb force in this situation, exerts zero torque about the point it acts towards (in this case, the positive charge +Q) because of the way torque is calculated and the properties of the central force.

Here’s the breakdown:

  • Torque (τ): This is the rotational equivalent of force. It’s calculated as the cross product of the position vector (r) of the point relative to the rotation axis and the force vector (F) acting on that point: τ = r x F.
  • Central Force: A central force always acts along the line connecting the center (positive charge +Q in this case) and the object it acts on (negative charge -q).
  • Zero Torque: When the force vector (F) is perfectly aligned with the position vector (r) or anti-aligned with it (180° apart), the cross product (r x F) becomes zero. This makes the torque acting on the object due to the central force zero.

In simpler terms, since the central force always acts directly towards the center (+Q), it has no twisting effect about that center. It only tries to pull the object (charge -q) straight in or out. This is why the torque is zero.

This property of central forces leads to the conservation of angular momentum in such systems. Without any external torque to change it, the total angular momentum of the system remains constant throughout its motion.