A conducting rod of mass \(m\) and length \(\ell\) slides along horizontal, frictionless conducting rails connected by a resistor of resistance \(R\). A uniform magnetic field of magnitude \(B\) is directed perpendicular to the plane of the rails. An external agent applies a force parallel to the rails such that the position of the rod as a function of time \(t\) is given by \(x(t) = C t^2\), where \(C\) is a positive constant and \(x(0) = 0\). Which of the following expressions gives the total mechanical work done by the external force on the rod from \(t = 0\) to \(t = T\)?