Four point charges +Q , -2Q , +3Q , and -Q are fixed at the corners A, B, C, and D respectively of a square of side a , as shown in the figure. What is the work done by an external agent in bringing a test charge q₀ from infinity to the center O of the square? (where ₀ is the permittivity of free space)

Step-by-step solution

The distance of the center O from each corner of the square is r = a 2 . The electric potential at the center O due to the four point charges is the algebraic sum of the potentials due to each individual charge: V_O = 1 4 ₀ ( Q_A r + Q_B r + Q_C r + Q_D r ) Substituting the given values of the charges: V_O = 1 4 ₀ r (Q - 2Q + 3Q - Q) V_O = 1 4 ₀ r (Q) Substituting r = a 2 : V_O = 1 4 ₀ ( a 2 ) Q = 2 Q 4 ₀ a The work done by an external agent in bringing a test charge q₀ from infinity to the center O is given by: W