The values of a and b for which the force F = (axy + z³) î + \( \,{x^2}\hat j + bx{z^2}\hat k\) is conservative are

Concept:

A conservative force is a force done in moving a particle from one point to another, such that the force is independent of the path taken by the particle. It depends only on the initial and final position of the particle.

Calculation:

For conservative force

∇ × F = 0

\(\begin{vmatrix} i & j & k\\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ axy+z^3 & x^2 & bxz^2 \end{vmatrix}\) = 0

⇒ i(0 - 0) - y(bz² - 3z²) + z(2x - ax) = 0 

⇒ b - 3 = 0 and 2 - a = 0

∴ a = 2, b = 3

The correct answer is option (1).