Question

Two points P and Q are maintained at the potentials of $ 10{\text{V}} $ and $ {\text{ - 4V}} $ respectively. The work done in moving $ 100 $ electrons from P to Q is:
(A) $ - 9.60 \times {10^{ - 17}}J $
(B) $ 9.60 \times {10^{ - 17}}J $
(C) $ - 2.24 \times {10^{ - 16}}J $
(D) $ 2.24 \times {10^{ - 16}}J $

Answer 1

Hint
To solve this question, we need to calculate the change in potential energy accompanying the movement of the electrons. For that we have to calculate the total charge of the electrons which must be put in the expression for the change in potential energy.
Formula Used: The formulae used in solving this question are given by
 $\Rightarrow Q = ne $, here $ Q $ is the total charge on $ n $ electrons, and $ e $ is the charge of each electron.
 $\Rightarrow U = qV $, here $ U $ is the potential energy of a charge $ q $ at a point where the electrostatic potential is $ V $

Complete step by step answer
We know that the charge of $ n $ electrons is given by
 $\Rightarrow Q = ne $
Now, according to the question we have $ n = 100 $. Also we know that $ e = - 1.6 \times {10^{ - 19}}C $. So the total charge of $ 100 $ electrons is
 $\Rightarrow Q = 100 \times \left( { - 1.6 \times {{10}^{ - 19}}} \right) $
 $\Rightarrow Q = - 1.6 \times {10^{ - 17}}C $ …...(1)
Now, these electrons are moved in an electrostatic field from the point P to the point Q. So the work done required to move these electrons will be equal to the difference in the potential energies at these two points, that is
 $\Rightarrow W = \Delta U $
 $\Rightarrow W = {U_2} - {U_1} $ …….(2)
Now, as the potential energy is given as
 $\Rightarrow U = qV $
So we have
 $\Rightarrow {U_1} = Q{V_1} $, and
 $\Rightarrow {U_2} = Q{V_2} $
So from (2) we get
 $\Rightarrow W = Q{V_2} - Q{V_1} $
 $\Rightarrow W = Q\left( {{V_2} - {V_1}} \right) $ ……..(3)
According to the question, we have
 $\Rightarrow {V_1} = 10{\text{V}} $
 $\Rightarrow {V_2} = - 4{\text{V}} $
Also from (1) we have
 $\Rightarrow Q = - 1.6 \times {10^{ - 17}}C $
Substituting these in (3) we get
 $\Rightarrow W = - 1.6 \times {10^{ - 17}}\left( { - 4 - 10} \right) $
 $\Rightarrow W = 22.4 \times {10^{ - 17}}J $
Writing in scientific notation, we finally get
 $\Rightarrow W = 2.24 \times {10^{ - 16}}J $
Thus the work done comes out to be equal to $ 2.24 \times {10^{ - 16}}J $
Hence, the correct answer Is option D.

Note
In this question, we don’t have any information regarding the velocity of the charges at any point. So, we have assumed that the charge is moving with the constant velocity and hence there is no change in kinetic energy involved. So there should be no confusion regarding the expression of work done which we have used in this solution.