Two charges +q and −q are separated by a distance d. Find an expression for the potential difference VBA = VB − VA, where points A and B are both a distance b from the charges.

LOL
zero, cancels
To find the potential difference VBA between points A and B, we can use the principle of superposition. The potential difference VBA is the difference in potential energy experienced by a unit positive test charge when moved from point A to point B.
The potential difference from point A to the positive charge +q can be calculated using the formula:
VAP = k * (q / r1)
Where:
- VAP is the potential difference between point A and the positive charge +q,
- k is the electrostatic constant (approximately equal to 9 x 10^9 Nm^2/C^2),
- q is the magnitude of the charge (+q or -q),
- r1 is the distance between point A and the positive charge +q.
Similarly, the potential difference from point A to the negative charge -q can be calculated using the formula:
VAN = -k * (q / r2)
Where:
- VAN is the potential difference between point A and the negative charge -q,
- k is the electrostatic constant (approximately equal to 9 x 10^9 Nm^2/C^2),
- q is the magnitude of the charge (+q or -q),
- r2 is the distance between point A and the negative charge -q.
Since the potential difference due to the positive charge is positive (VAP), and the potential difference due to the negative charge is negative (VAN), we can express the potential difference VBA as the sum of these two potentials:
VBA = VAP + VAN
To find the potential difference between points B and the charges, we use similar formulas for VBP (potential difference between point B and the positive charge +q) and VBN (potential difference between point B and the negative charge -q):
VBP = k * (q / r3)
Where:
- VBP is the potential difference between point B and the positive charge +q,
- k is the electrostatic constant (approximately equal to 9 x 10^9 Nm^2/C^2),
- q is the magnitude of the charge (+q or -q),
- r3 is the distance between point B and the positive charge +q.
VBN = -k * (q / r4)
Where:
- VBN is the potential difference between point B and the negative charge -q,
- k is the electrostatic constant (approximately equal to 9 x 10^9 Nm^2/C^2),
- q is the magnitude of the charge (+q or -q),
- r4 is the distance between point B and the negative charge -q.
Similarly, we can express the potential difference between points A and B as the sum of these two potentials:
VBA = VBP + VBN
Since we are given that points A and B are equidistant from the charges, i.e., r1 = r2 = r3 = r4 = b, we can substitute the values of VAP, VAN, VBP, and VBN into the expression for VBA:
VBA = (k * q / b) - (k * q / b)
= 0
Therefore, the potential difference VBA between points A and B, when they are both at a distance b from the charges, is zero.