Two plates separated by a distance d = 14.8 mm are charged to a potential difference V = 7.25 V . A constant force F = 7.81 N pushes a − 8.30 mC charge from the positively charged plate to the negatively charged plate. Calculate the change in the kinetic energy Δ K , potential energy Δ U , and total energy Δ E of the charge as it travels from one plate to the other. Assume the initial speed of the charge is 0.
The Correct Answer and Explanation is:
Given Data:
- Distance between plates, ( d = 14.8 \, \text{mm} = 14.8 \times 10^{-3} \, \text{m} )
- Potential difference between plates, ( V = 7.25 \, \text{V} )
- Constant force acting on the charge, ( F = 7.81 \, \text{N} )
- Magnitude of the charge, ( q = -8.30 \, \text{mC} = -8.30 \times 10^{-3} \, \text{C} )
- Initial velocity, ( v_0 = 0 )
We need to calculate:
- Change in kinetic energy, ( \Delta K )
- Change in potential energy, ( \Delta U )
- Change in total energy, ( \Delta E )
Step 1: Calculating the Work Done
The work done by the force ( F ) in moving the charge from one plate to the other is given by:
[
W = F \times d
]
Substituting the values:
[
W = 7.81 \, \text{N} \times 14.8 \times 10^{-3} \, \text{m} = 0.115 \, \text{J}
]
This work done is transferred into the change in kinetic energy, as the force is responsible for the movement of the charge, assuming no friction or other forces acting on it.
Thus, the change in kinetic energy is:
[
\Delta K = W = 0.115 \, \text{J}
]
Step 2: Calculating the Change in Potential Energy
The potential energy associated with the charge due to the potential difference between the plates is given by:
[
\Delta U = q \times V
]
Substituting the values:
[
\Delta U = (-8.30 \times 10^{-3} \, \text{C}) \times (7.25 \, \text{V}) = -0.060225 \, \text{J}
]
Thus, the change in potential energy is:
[
\Delta U = -0.0602 \, \text{J}
]
Step 3: Total Energy Change
The total energy change ( \Delta E ) is the sum of the changes in kinetic and potential energies:
[
\Delta E = \Delta K + \Delta U
]
Substituting the values:
[
\Delta E = 0.115 \, \text{J} + (-0.0602 \, \text{J}) = 0.0548 \, \text{J}
]
Final Results:
- Change in kinetic energy, ( \Delta K = 0.115 \, \text{J} )
- Change in potential energy, ( \Delta U = -0.0602 \, \text{J} )
- Change in total energy, ( \Delta E = 0.0548 \, \text{J} )
Explanation:
The work done by the constant force ( F ) in moving the charge from the positively charged plate to the negatively charged plate directly translates into a change in kinetic energy, since the force accelerates the charge. The potential energy change is computed using the charge’s interaction with the electric field created by the potential difference between the plates. Since the charge is negative, the potential energy decreases as it moves toward the negatively charged plate. Finally, the total energy change reflects the net effect of the kinetic and potential energy changes.