A 10.0 microCoulomb point charge is fixed in place at the origin. A 2.00-microCoulomb point charge is released from rest at x = 0.500 m. Because of the repulsive force between the two point charges, the 2.00-microCoulomb charge is accelerated in the +x direction. What is the kinetic energy of the 2.00-microCoulomb point charge when it is at x = 0.800 m?
To find the kinetic energy of the 2.00-microCoulomb point charge when it is at x = 0.800 m, we need to calculate the work done on the charge as it moves from x = 0.500 m to x = 0.800 m, and then use the definition of kinetic energy.
The work done on the charge can be calculated using the formula:
Work = Change in Potential Energy
The potential energy of a point charge in an electrostatic field can be calculated using the formula:
Potential Energy = (k * q1 * q2) / r
Where:
- k is Coulomb's constant (9.0 x 10^9 N * m^2 / C^2)
- q1 and q2 are the charges of the two point charges (in Coulombs)
- r is the distance between the two charges (in meters)
In this case, the fixed charge at the origin (q1) is 10.0 microCoulombs and the moving charge (q2) is 2.00 microCoulombs. The distance between them is the difference between the final position (0.800 m) and the initial position (0.500 m), which is 0.300 m.
Using these values, we can calculate the potential energy at the initial position and final position, and then find the work done:
Potential Energy_initial = (9.0 x 10^9 N * m^2 / C^2) * (10.0 x 10^-6 C) * (2.00 x 10^-6 C) / 0.500 m
Potential Energy_final = (9.0 x 10^9 N * m^2 / C^2) * (10.0 x 10^-6 C) * (2.00 x 10^-6 C) / 0.800 m
The work done on the charge is the difference between the potential energy at the final position and the initial position:
Work = Potential Energy_initial - Potential Energy_final
Finally, we can use the definition of kinetic energy to find the kinetic energy of the point charge when it is at x = 0.800 m:
Kinetic Energy = Work
(Alternatively, we can use the equation Kinetic Energy = 1/2 * m * v^2, if we know the mass of the charge and its velocity, but in this case, we only have information about the charges and their positions.)
By calculating the work done using the formulas and values provided, we can find the kinetic energy of the point charge when it is at x = 0.800 m.