A proton is moving toward a charge of + 2.8 nC. On its way in to the charge it passes Point B at 4.5 cm from the charge and then passes Point A at 1.2 cm as it gets closer to the charge.
a. What is the potential difference between points B and A?
b. What is the change in potential energy between points B and A?
c. If the speed of the proton at point B was 5.8x105 m/s, what is its speed at point A?
To find the potential difference between points B and A (VBA), we can use the formula:
VBA = VB - VA
where VB is the potential at point B and VA is the potential at point A.
The potential at a distance r from a point charge q can be calculated using the formula:
VB = k * (q / r)
where k is the electrostatic constant, equal to 9 × 10^9 N m^2/C^2. In this case, q is the charge of +2.8 nC (or +2.8 × 10^-9 C) and r is the distance from the charge to point B (4.5 cm or 0.045 m). Plugging in these values, we get:
VB = (9 × 10^9 N m^2/C^2) * (+2.8 × 10^-9 C) / (0.045 m)
Solving this equation gives us the value of VB.
Similarly, we can calculate the potential at point A (VA) by substituting r = 0.012 m into the formula:
VA = (9 × 10^9 N m^2/C^2) * (+2.8 × 10^-9 C) / (0.012 m)
Once we have the values of VB and VA, we can find VBA by subtracting VA from VB.
To find the change in potential energy between points B and A (ΔU), we can use the formula:
ΔU = q * (VA - VB)
where q is the charge of the proton and VA and VB are the potentials at points A and B, respectively.
Using the previously calculated values of q (which is +2.8 × 10^-9 C) and the potentials at points A and B, we can find the change in potential energy.
Finally, to find the speed of the proton at point A, we need to apply the conservation of mechanical energy. The mechanical energy of the proton is the sum of its potential energy and kinetic energy:
E = U + K
where E is the mechanical energy, U is the potential energy, and K is the kinetic energy.
Since the charge of the proton does not change during its motion, the change in potential energy (ΔU) will be equal to the negative of the change in kinetic energy (ΔK). Therefore:
ΔU = -ΔK
Using the previously calculated value of ΔU, we can find the change in kinetic energy. We can then subtract the change in kinetic energy from the initial kinetic energy at point B to find the final kinetic energy at point A. From the kinetic energy, we can determine the speed of the proton at point A using the formula:
K = (1/2) * m * v^2
where m is the mass of the proton and v is its velocity. By substituting the values of m and K, we can solve for v, which will give us the speed of the proton at point A.