An ideal, incompressible fluid with density \(\rho\) flows through a horizontal, cylindrical pipe. In the wide section of the pipe, the fluid has a cross-sectional area \(A_1\), a pressure \(P_1\), and flows with a speed \(v_1\). The pipe then narrows to a section with a cross-sectional area \(A_2 = \dfrac{1}{2}A_1\), where the pressure is \(P_2\) and the speed is \(v_2\).

Which of the following is a correct expression for the pressure difference \(P_1 - P_2\) in terms of \(\rho\) and \(v_1\)?