The electric potential V along the x -axis in a region is given by V(x) = x^3 - 3x (in volts), where x is in meters. A particle with a positive charge +q is constrained to move along the x -axis. The x -coordinate where the particle will be in stable equilibrium is :

Step-by-step solution

The potential energy of the particle is U(x) = qV(x) = q(x^3 - 3x) . For equilibrium, the net force on the particle must be zero. The force is given by F = - dU dx . F = -q d dx (x^3 - 3x) = -q(3x^2 - 3) Setting F = 0 for equilibrium: 3x^2 - 3 = 0 x^2 = 1 x = 1 m or x = -1 m For stable equilibrium, the potential energy must be a local minimum, which requires d^2U dx^2 > 0 . d^2U dx^2 = d dx [q(3x^2 - 3)] = q(6x) At x = 1 m , d^2U dx^2 = 6q > 0 (since q is positive). This corresponds to stable equilibrium. At x = -1