Q3  Given in Fig. 6.11 are examples of some potential energy functions in one dimension. The total energy of the particle is indicated by a cross on the ordinate axis. In each case, specify the regions, if any, in which the particle cannot be found for the given energy. Also, indicate the minimum total energy the particle must have  in each  case. Think of simple physical contexts for which these potential energy shapes are relevant.

          

 

Answers (1)

Total energy = kinetic energy (KE) + potential energy(PE)

KE > 0 since m and v2 is positive. If KE <0 particles cannot be find. If PE>TE, then KE<0 (now in all graph check for this condition)

In case 1 kinetic energy is negative for x<a. So at x<a particle cannot be found.

In case 2 for x<a and for x> b kinetic energy is negative. So the particle cannot be found in these regions.

In the third case, the minimum potential energy is when   a<x<b. At this position, the potential energy is negative (- V1).   

The kinetic energy in this case is given by :

                                                           K.E.\ =\ E\ -\ (-V_1)\ =\ E\ +\ V_1

And the minimum energy of particle is  - V1.

 

In the fourth case,  the particle will not exist in the states which will have potential energy greater than the total energy.

Thus particle will not exist in    \frac{-b}{2}<x<\frac{b}{2}   and   \frac{-a}{2}<x<\frac{a}{2} .

The minimum energy of particle will be - V1 as it is the minimum potential energy.

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