Let's determine the kinetic energy gained by the particle with charge [tex]\( q \)[/tex] as it moves in the presence of a fixed charge [tex]\( Q \)[/tex].
To solve this question, we need to follow these steps:
1. Understand the Given Data:
- Charge of the moving particle [tex]\( q = +15.0 \mu \text{C} \)[/tex].
- Charge of the fixed particle [tex]\( Q = +60.0 \mu \text{C} \)[/tex].
- Initial position of the moving particle [tex]\( x_i = 0.80 \text{ meters} \)[/tex].
- Final position of the moving particle [tex]\( x_f = 1.20 \text{ meters} \)[/tex].
- Coulomb's constant [tex]\( k = 8.99 \times 10^9 \, \text{N·m}^2/\text{C}^2 \)[/tex].
2. Convert Charges to Coulombs:
- [tex]\( q = 15.0 \times 10^{-6} \, \text{C} \)[/tex].
- [tex]\( Q = 60.0 \times 10^{-6} \, \text{C} \)[/tex].
3. Calculate Initial Electric Potential Energy:
The initial electric potential energy [tex]\( U_i \)[/tex] at [tex]\( x_i = 0.80 \text{ m} \)[/tex] can be calculated using the formula for electrical potential energy:
[tex]\[
U_i = \frac{k \cdot q \cdot Q}{r_i}
\][/tex]
Substituting in the given values:
[tex]\[
U_i = \frac{8.99 \times 10^9 \, \text{N·m}^2/\text{C}^2 \cdot 15.0 \times 10^{-6} \, \text{C} \cdot 60.0 \times 10^{-6} \, \text{C}}{0.80 \, \text{m}}
\][/tex]
[tex]\[
U_i \approx 10.114 \, \text{J}
\][/tex]
4. Calculate Final Electric Potential Energy:
The final electric potential energy [tex]\( U_f \)[/tex] at [tex]\( x_f = 1.20 \text{ m} \)[/tex] can be calculated by:
[tex]\[
U_f = \frac{k \cdot q \cdot Q}{r_f}
\][/tex]
Substituting in the given values:
[tex]\[
U_f = \frac{8.99 \times 10^9 \, \text{N·m}^2/\text{C}^2 \cdot 15.0 \times 10^{-6} \, \text{C} \cdot 60.0 \times 10^{-6} \, \text{C}}{1.20 \, \text{m}}
\][/tex]
[tex]\[
U_f \approx 6.743 \, \text{J}
\][/tex]
5. Determine the Kinetic Energy:
The kinetic energy of the particle as it moves from the initial position to the final position is gained by the loss of electric potential energy. Therefore, the kinetic energy ([tex]\( K \)[/tex]) can be calculated as:
[tex]\[
K = U_i - U_f
\][/tex]
Substituting in the obtained potential energies:
[tex]\[
K = 10.114 \, \text{J} - 6.743 \, \text{J}
\][/tex]
[tex]\[
K \approx 3.371 \, \text{J}
\][/tex]
Therefore, the kinetic energy of the particle with charge [tex]\( q \)[/tex] at the instant it has moved to a point [tex]\( x = 1.20 \text{ meters} \)[/tex] on the x-axis is approximately [tex]\( 3.371 \text{ Joules} \)[/tex].
