Answer:
a. d/2 mid-way between the charges.
b. d/2 mid-way between the charges.
Explanation:
(a) Find the location of all points, if any, where the electric potential is zero.
Since the charges are of equal magnitude and opposite charge and separated by a distance, d, the electric potential due to the +Q charge is V = kQ/x and that due to the -Q charge is V' = -kQ/(d - x) where x is the point of zero electric potential.
The potential is zero when V + V' = 0, and this can only be midway between the charges. This is shown below
So, kQ/x + [-kQ/(d - x)] = 0
kQ/x - kQ/(d - x) = 0
kQ/x = kQ/(d - x)
1/x = 1/(d - x)
(d - x) = x
d = x + x
d = 2x
x = d/2 which is mid-way between the charges.
(b) Find the location of all points, if any, where the electric field is zero.
Since the charges are of equal magnitude and opposite charge and separated by a distance, d, the electric field due to the +Q charge is E = kQ/x² and that due to the -Q charge is E' = -kQ/(d - x)² where x is the point of zero electric field.
The electric field is zero when E + E' = 0 and this can only be midway between the charges. This is shown below.
So, kQ/x² + [-kQ/(d - x)²] = 0
kQ/x² - kQ/(d - x)² = 0
kQ/x² = kQ/(d - x)²
1/x² = 1/(d - x)²
(d - x)² = x²
d - x = ± x
d = x ± x
d = x - x or x + x
d = 0 or 2x
d = 0 or d = 2x
Since d ≠ 0, d = 2x ⇒ x = d/2 which is midway between the charges.
