Question

A particle with a charge of \(+5.00 \mu \mathrm{C}\) is released from rest at a point on the \(x\) -axis, where \(x=0.100 \mathrm{~m} .\) It begins to move as a result of the presence of a \(+9.00-\mu C\) charge that remains fixed at the origin. What is the kinetic energy of the particle at the instant it passes the point \(x=0.200 \mathrm{~m} ?\)

Short answer

#tag_title# Summary When we calculated the final kinetic energy, we found a negative value, which is not possible for kinetic energy. This inconsistency means the charged particle at x=0.100m cannot move towards x=0.200m, as the repulsive force between both positive charges will push the moving particle away from the fixed charge at the origin.

Step-by-step solution

Calculate the initial potential energy

To find the initial potential energy of the system, we need to use the formula for potential energy due to Coulomb's law: \(U_{initial} = k \frac{Q_1 Q_2}{r_{initial}}\) where \(k = 8.99 \times 10^9 N m^2 C^{-2}\) (Coulomb's constant), \(Q_1 = +5.00 \times 10^{-6} C\), \(Q_2 = +9.00 \times 10^{-6} C\), and \(r_{initial} = 0.100 m\). So, \(U_{initial} = (8.99 \times 10^9) \frac{(5.00 \times 10^{-6})(9.00 \times 10^{-6})}{0.100} = 4.05 \times 10^{-4} J\)

Calculate the final potential energy

Now, we will calculate the final potential energy when the moving particle reaches x = 0.200 m. The formula for potential energy remains the same: \(U_{final} = k \frac{Q_1 Q_2}{r_{final}}\) where \(r_{final} = 0.200 m\). So, \(U_{final} = (8.99 \times 10^9) \frac{(5.00 \times 10^{-6})(9.00 \times 10^{-6})}{0.200} = 2.025 \times 10^{-4} J\)

Find the kinetic energy using the conservation of energy principle

According to the conservation of energy principle, the total energy of the system remains constant. Therefore, the change in potential energy (\(\Delta U = U_{final} - U_{initial}\)) should be equal to the change in kinetic energy (\(\Delta K\)). \(\Delta U = \Delta K\) Since the moving particle is initially at rest, its initial kinetic energy is zero. Therefore, the final kinetic energy (\(K_{final}\)) is the same as the change in kinetic energy: \(K_{final} = U_{final} - U_{initial}\) \(K_{final} = 2.025 \times 10^{-4} J - 4.05 \times 10^{-4} J = -2.025 \times 10^{-4} J\) However, kinetic energy cannot be negative. The negative sign indicates that the problem description is inconsistent, meaning that the charged particle at x=0.100m cannot move towards x=0.200m as the repulsive force between both positive charges will push the moving particle away from the fixed charge at the origin.