A charge of +4q is fixed to one corner of a square, while a charge of -9q is fixed to the opposite corner. Expressed in terms of q, what charge should be fixed to the center of the square, so the potential is zero at each of the two empty corners?
To find the charge that should be fixed at the center of the square to make the potential zero at each of the two empty corners, we can use the principle of superposition.
The potential at a point in space due to multiple charges is equal to the sum of the potentials due to each individual charge. Additionally, the potential due to a point charge can be calculated using the formula:
V = k * (q / r)
Where V is the potential, k is the electrostatic constant (k = 9 x 10^9 Nm^2/C^2), q is the charge, and r is the distance from the point charge.
Let's consider the charge at the center of the square. The potential at an empty corner due to this center charge should be zero. Since both corners are equidistant from the center of the square, their potentials will have the same magnitude but opposite signs when the potential at each corner is zero.
Now, let's calculate the potential at one of the empty corners due to the center charge. Since the two charges at the corners are fixed, the center charge will only affect the potential at the empty corners.
The distance between the center charge and the empty corner is the diagonal length of the square, which can be found using the Pythagorean theorem. The length of one side of the square is the distance between the two fixed charges, which is a diagonal of the square. Let's call this length "d":
d^2 = (side length of square)^2 + (side length of square)^2
d^2 = 2(side length of square)^2
d = √2 x (side length of square)
Now, let's calculate the potential at the empty corner due to the center charge:
V_empty_corner = k * (q_center_charge / r_empty_corner)
Since we want the potential at the empty corner to be zero, we can set:
0 = k * (q_center_charge / r_empty_corner)
Simplifying, we have:
0 = k * (q_center_charge / (√2 x side length of square))
To cancel out the k and (√2 x side length of square) terms, we can multiply both sides of the equation by (√2 x side length of square):
0 * (√2 x side length of square) = k * q_center_charge
Dividing both sides by k, we get:
0 = q_center_charge / k
So, the charge at the center of the square, expressed in terms of q, should be zero. Therefore, no charge needs to be fixed at the center of the square to make the potential zero at each of the two empty corners.