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.
To find the potential difference (VB - VA) between points A and B, we need to calculate the work done by the electric force in moving the particle from A to B.
The work done by a force is given by the equation:
Work (W) = Force (F) x Distance (d) x cos(θ)
In this case, the force acting on the particle is the electric force. The electric force (Fe) experienced by a charged particle is given by:
Fe = q * E
where q is the charge of the particle and E is the electric field strength.
As the particle is moving from A to B, the force and displacement vectors are parallel, so the angle between them, θ, is 0. Therefore, the equation for work becomes:
Work (W) = q * E * d
We can rewrite the equation in terms of potential difference (V) using the formula:
Potential Difference = Work / Charge
V = W / q
Given that the particle has a charge of -3.2 μC and the only force acting on it is the electric force, we can substitute these values into the equation:
V = W / q = (Fe * d) / q
First, let's find the electric field strength (E) at point A. The electric force (Fe) experienced by the particle is equal to the product of its charge and the electric field strength:
Fe = q * E
Rearranging the equation, we can solve for E:
E = Fe / q
At point A, the particle is at rest, which means it has zero initial kinetic energy. Therefore, all the work done by the electric force goes into increasing the particle's kinetic energy.
The work done (W) by the electric force is equal to the change in kinetic energy (ΔK) of the particle. We can calculate ΔK using the equation:
ΔK = (1/2) * m * (vf^2 - vi^2)
where m is the mass of the particle, vf is the final velocity (58 m/s), and vi is the initial velocity (zero as the particle is at rest).
Plugging in the values, we get:
ΔK = (1/2) * (4.0 x 10^-6 kg) * (58 m/s)^2
Next, we can calculate the electric field strength (E) at point A:
E = Fe / q = (ΔK / d) / q
d is the distance traveled by the particle. However, the problem does not provide information about the distance between A and B. Hence, we cannot determine the exact value of the potential difference (VB - VA) between A and B without this distance.
Therefore, without the distance (d) information, we cannot calculate the potential difference (VB - VA) between points A and B.