The potential energy at a point, relative to the reference point, is defined as the negative of the work done by the conservative force as the object moves from the reference point to the point consid

Question

The potential energy at a point, relative to the reference point, is defined as the negative of the work done by the conservative force as the object moves from the reference point to the point considered. The value of potential energy at the reference point itself can be set equal to zero because we are always concerned only with differences of potential energy between two points and the associated change of kinetic energy. A particle A is fixed at the origin of a fixed coordinate system. Another particle B, which is free to move, experiences a force $F = (- \frac{2 α}{r ^{3}} + \frac{β}{r ^{2}}) \overset{r}{^}$ due to particle A, where $r$ is the position vector of particle B relative to A. It is given that the force is conservative in nature and the potential energy at infinity is zero. If B has to be removed from the influence of A, energy has to be supplied for such a process. The ionization energy E₀ is the work that has to be done by an external agent to move the particle from a distance $r_{0}$ to infinity slowly. Here, $r_{0}$ is the equilibrium position of the particle.

Find the ionization energy $E_{0}$ of the particle B .

TestHub · Accepted Answer

F = 0 ⇒ − r 0 3 ​ 2 α ​ + r 0 2 ​ β ​ = 0 ⇒ r 0 ​ = β 2 α ​ E 0 ​ = − Δ U = − ( U ∞ ​ − U r 0 ​ ​ ) ⇒ [ 0 − ( − r 0 2 ​ α ​ + r 0 ​ β ​ ) ] E 0 ​ = ( 2 α / β ) 2 α ​ − ( 2 α / β ) β ​ ⇒ 4 α β 2 ​ − 2 α β 2 ​ = − 4 α β 2 ​ = ∣ E 0 ​ ∣ = 4 α β 2 ​

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