To calculate the minimum work required to move the point charge q from point B to A, we need to calculate the electric potential difference between the two points, and then use the equation W = qΔV, where W is the work done, q is the charge being moved, and ΔV is the potential difference.
To find the electric potential at point A and B due to the charged ring, we can use the equation for electric potential due to a charged ring:
V = kQ/r
Where k is Coulomb's constant, Q is the total charge of the ring, and r is the distance from the center of the ring to the point where the potential is being calculated.
For point B, the potential due to the charged ring is:
VB = kQ/r = (8.99 × 10^9 N·m^2/C^2) * (-820 × 10^-9 C) / (2.4 m) = -306.55 V
For point A, the potential due to the charged ring is:
VA = kQ/r = (8.99 × 10^9 N·m^2/C^2) * (-820 × 10^-9 C) / (4.8 m) = -153.27 V
The potential difference between point A and B is:
ΔV = VA - VB = (-153.27 V) - (-306.55 V) = 153.28 V
The minimum work required to move the charge q from point B to A is:
W = qΔV = (530 × 10^-9 C) * (153.28 V) = 81.09 × 10^-6 J
Therefore, the minimum work required to transfer the electron from B to A is 81.09 × 10^-6 J.