two point charges +q and -q are separated by a distance d. aside from infinity, where else is the potential due to these two charges 0?
midway between the charges, of course.
To find the locations other than infinity where the potential due to the two charges is zero, we need to consider the concept of the electric potential due to point charges.
The electric potential due to a point charge can be computed using the formula:
V = k * (Q / r),
where V is the electric potential, k is the Coulomb's constant (k = 9 x 10^9 Nm^2/C^2), Q is the magnitude of the point charge, and r is the distance from the charge.
Since we have two charges (+q and -q) separated by a distance d, we need to find the locations where the sum of the electric potentials due to these charges is zero.
Let's consider a point P at a distance x from the positive charge +q. The distance from the negative charge -q would be (d - x), as the total distance between the charges is d.
The electric potential at point P due to the positive charge is:
V1 = k * (q / x),
while the electric potential at the same point due to the negative charge is:
V2 = k * (-q / (d - x)).
Setting the sum of these two potentials to zero gives:
V1 + V2 = 0,
k * (q / x) + k * (-q / (d - x)) = 0.
To simplify the equation, we can multiply through by x(d - x) to eliminate the denominators:
q * (d - x) - q * x = 0,
qd - qx - qx = 0,
qd - 2qx = 0.
Now, solve the equation for x:
qd = 2qx,
x = (qd) / (2q).
From this equation, we can see that x depends on the distance between the charges (d) and the magnitude of the charge (+q or -q). If we set x equal to zero, we recover one of the charges. However, if we set x equal to d, we get the other charge. Therefore, the only other point where the potential due to these two charges is zero (aside from infinity) is at the midpoint between the charges:
x = d/2.
So, the potential due to these two charges is zero at the midpoint between them, which is at a distance of d/2 from each charge.