For Question 44
Considering that the loop is complete, we know that the electric field at the center will be zero. Now consider an element on the ring of length ∂l. The electric field exerted by this element at the loop center must be equal in magnitude and opposite in direction to the field exerted by the rest of the loop at the its center. So
E(at center due to the remaining wire) = E(at center due to ∂l)


Which gives the desired answer.