Suppose a charge q is placed at point x = 0, y = 0. A second charge q is placed at point x = 6.2 m, y = 0. What charge Q must be placed at the point x = 3.1 m, y = 0 in order that the field at the point x = 3.1 m, y = 4.2 m be zero?
To find the charge Q that must be placed at the point (x = 3.1 m, y = 0) in order for the electric field at the point (x = 3.1 m, y = 4.2 m) to be zero, we need to use the concept of electrostatic force and the superposition principle.
The electric field at a point in space is determined by the electric forces exerted by all the charges in the vicinity. For an electric field to be zero at a particular point, the electric forces from all the charges must cancel each other out.
In this case, we have two point charges q placed at (x = 0, y = 0) and (x = 6.2 m, y = 0), and we want to find the charge Q that must be placed at (x = 3.1 m, y = 0) so that the electric field at (x = 3.1 m, y = 4.2 m) is zero.
First, let's calculate the electric field at (x = 3.1 m, y = 4.2 m) due to the charge q at (x = 0, y = 0) and the charge Q at (x = 3.1 m, y = 0). We will denote the electric field as E1.
E1 = k * (q / r^2) - k * (Q / r^2)
Here, k is Coulomb's constant, r is the distance between the charges and the point of interest (in this case, (x = 3.1 m, y = 4.2 m)).
Next, let's calculate the electric field at (x = 3.1 m, y = 4.2 m) due to the charge q at (x = 6.2 m, y = 0). We will denote the electric field as E2.
E2 = k * (q / r^2)
To cancel out the electric field E2, the electric field E1 must be equal in magnitude but opposite in direction. This means:
E1 = -E2
Substituting the expressions for E1 and E2, we have:
k * (q / r^2) - k * (Q / r^2) = -k * (q / r^2)
Cancelling out the common factors and rearranging the equation, we get:
Q = -q
Therefore, for the electric field at (x = 3.1 m, y = 4.2 m) to be zero, the charge Q at (x = 3.1 m, y = 0) must be equal in magnitude but opposite in sign to the charge q at (x = 0, y = 0).