Two charges, +Q and −Q, have been placed on the x-axis at x=−a and x=a respectively.
a) Where will the electric field be zero?
b) Where will the electric potential be zero?
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To find the locations where the electric field is zero, we need to identify the points on the x-axis where the electric fields produced by the two charges cancel each other out. The electric field produced by a point charge is given by Coulomb's law:
E = k * Q / r^2
Where E is the electric field, k is the electrostatic constant (9 x 10^9 Nm^2/C^2), Q is the charge, and r is the distance from the charge.
a) To determine the points on the x-axis where the electric field is zero, we set up an equation that equates the magnitudes of the electric fields produced by the two charges:
k * Q / (r1^2) = k * (-Q) / (r2^2)
Simplifying the equation, we get:
Q / r1^2 = -Q / r2^2
r2^2 / r1^2 = -1
Taking the square root of both sides:
(r2 / r1) = √(-1)
Since the square root of a negative number is not defined in real numbers, we can conclude that there are no points on the x-axis where the electric field is zero.
b) To find the locations where the electric potential is zero, we set up a similar equation. The electric potential at a point due to a point charge is given by:
V = k * Q / r
Where V is the electric potential.
By equating the electric potentials produced by the two charges, we get:
k * Q / r1 = k * (-Q) / r2
Simplifying the equation, we have:
Q / r1 = -Q / r2
r2 / r1 = -1
So, the ratio of the distances from the two charges is -1. This means that the point on the x-axis where the electric potential is zero is at the midpoint between the two charges, which is the origin (x = 0).