A particle with a charge of -3.2 μC and a mass of 4.0 x 10-6 kg is released from rest at point A and accelerates toward point B, arriving there with a speed of 58 m/s. The only force acting on the particle is the electric force. What is the potential difference VB - VA between A and B? If VB is greater than VA, then give the answer as a positive number. If VB is less than VA, then give the answer as a negative number.
Where would I start with this question?
To solve this question, we need to use the concept of electric potential energy and the work-energy principle.
The electric potential difference (V) between two points is equal to the change in electric potential energy (U) per unit charge. Mathematically, it can be represented as V = U/q, where U is the change in electric potential energy and q is the charge.
In this problem, we are given the charge of the particle (-3.2 μC) and the speed of the particle (58 m/s) at point B. We know that the electric force is the only force acting on the particle, and it is conservative. Therefore, the work done by the electric force can be converted into a change in potential energy.
To find the potential difference (VB - VA) between points A and B, we will follow these steps:
1. Calculate the change in potential energy (U) between A and B.
- The change in potential energy is given by the formula U = qV, where q is the charge and V is the potential difference.
- U = (charge) x (potential difference)
- U = (-3.2 μC) x (VB - VA)
2. Calculate the kinetic energy (K) of the particle at point B.
- The kinetic energy can be calculated as K = (1/2) mv^2, where m is the mass of the particle and v is the speed of the particle.
- K = (1/2) (4.0 x 10^-6 kg) (58 m/s)^2
3. Apply the work-energy principle.
- According to the work-energy principle, the work done on an object is equal to the change in its energy.
- In this case, the work done by the electric force is equal to the change in potential energy:
W = U
- The work done by the electric force can also be expressed as the change in kinetic energy:
W = Kf - Ki, where Kf is the final kinetic energy and Ki is the initial kinetic energy.
- Kf = 0 (since the particle ends at point B with zero speed)
- Ki = (1/2) (4.0 x 10^-6 kg) (0 m/s)^2 (since the particle starts at rest at point A)
- W = 0 - [(1/2) (4.0 x 10^-6 kg) (58 m/s)^2]
4. Set the work done by the electric force equal to the change in potential energy and solve for VB - VA.
- 0 - [(1/2) (4.0 x 10^-6 kg) (58 m/s)^2] = (-3.2 μC) x (VB - VA)
By solving this equation, you can find the potential difference VB - VA between points A and B.