Question

A uniformly charged ring of radius 3a and total charge q is placed in xy-plane centered at origin. A point charge q is moving towards the ring along the z-axis and has speed u at z−4a. What is the minimum value of u such that it crosses the origin?

Answer 1

Hint: Here, as we know according to the law of conservation of energy, final and initial energy will be the same. We will use this relation to find the required value of v, by applying the given values. So, the value of v will give the required result.

Formula used:
${U_i} + {K_{i}} = {U_{f}} + {K_f}$

Complete step-by-step answer:
The relation between the potential energy and kinetic energy is used. Since, we know that the total energy of a system is conserved; we have the final and the initial kinetic and potential energies of the particle:
$\eqalign{
  & {U_i} + {K_{i}} = {U_{f}} + {K_f} \cr
  \Rightarrow & \dfrac{{k{q^2}}}{{\sqrt {16a^2 + 9a^2} }} + \dfrac{1}{2}m{v^2} = \dfrac{{k{q^2}}}{{3a}} \cr
 \Rightarrow & \dfrac{{1}}{2}m{v^2} = k{q^2}(\dfrac{1}{3a} - ( \dfrac{{1}}{5a})) = \dfrac{{2k{q^2}}}{{15a}} \cr
  \Rightarrow & \therefore v = \sqrt {\dfrac{{4k{q^2}}}{{15ma}}} \cr} $
Therefore, the velocity of the particle on the ring of radius r is given by v.

Additional Information: The total energy of a system is the sum of kinetic energy and the potential energy. Also, we know that according to the law of conservation of energy, energy can neither be created nor be destroyed; it can only be transferred from one form to another. Electric energy is converted to heat energy by the use of a water heater.
 Potential energy is defined as the energy that is stored in an object due to its position relative to some zero position. An object or a body possesses gravitational potential energy if it is positioned at a height above or below the zero height.
Further, Kinetic energy is defined as the form of energy that an object or a particle has by reason of its motion. Kinetic energy is a property of a moving object or particle.

Note: Law of conservation of energy is needed to be remembered i.e., energy can neither be created nor be destroyed, it can only be transferred from one form to another. If a particle or body is only in motion but does not have any height, the potential energy in that case will be zero.