Final Answer:
(a) The electric potential at the free corner, where there is no charge, is zero.
(b) To achieve zero electric potential at the center of the rectangle, a charge of +1.5 µC should be placed at the free corner.
Step-by-step explanation:
In a system of point charges, the electric potential at a point due to multiple charges is the sum of the potentials created by each individual charge. At the free corner without any charge (q4 = 0), the electric potential is zero because there's no charge contributing to the potential at that point.
To find the charge needed at the free corner for the electric potential at the center of the rectangle to be zero, we consider the principle that at the center of a rectangle, the electric potential due to opposite charges at the ends of the diagonals should cancel out.
Here, q1 and q2 create equal and opposite potentials at the center. The potential due to q1 is positive and that due to q2 is negative. To balance these, a charge q4 must be introduced at the free corner, so the total potential becomes zero at the center.
Using the formula for electric potential
V= rk⋅∣q∣ , where
k is Coulomb's constant, ∣q∣ is the magnitude of charge, and r is the distance, we can calculate the charge needed at the free corner. Considering q1 and q2 at the center with a distance d between them, the potentials need to cancel out, giving
4=0V q1+V
q2 +V
q4=0. Solving for 4q4, we get 4=−1
⋅
2q4= q2−q1⋅d . Substituting the given values,
4=−9.00−9.00=1.5q4= −9.00μC
−9.00μC⋅d=1.5μC, with a positive sign indicating the required charge's direction.
Therefore, a positive charge of +1.5 µC placed at the free corner would create an electric potential of zero at the center of the rectangle, balancing the potentials from q1 and q2.