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Question: Two points P and Q are maintained at the potentials of \( 10{\text{V}} \) and \( {\text{ - 4V}} \) r...

Two points P and Q are maintained at the potentials of 10V10{\text{V}} and  - 4V{\text{ - 4V}} respectively. The work done in moving 100100 electrons from P to Q is:
(A) −9.60×10−17J- 9.60 \times {10^{ - 17}}J
(B) 9.60×10−17J9.60 \times {10^{ - 17}}J
(C) −2.24×10−16J- 2.24 \times {10^{ - 16}}J
(D) 2.24×10−16J2.24 \times {10^{ - 16}}J

Explanation

Solution

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
⇒Q=ne\Rightarrow Q = ne , here QQ is the total charge on nn electrons, and ee is the charge of each electron.
⇒U=qV\Rightarrow U = qV , here UU is the potential energy of a charge qq at a point where the electrostatic potential is VV

Complete step by step answer
We know that the charge of nn electrons is given by
⇒Q=ne\Rightarrow Q = ne
Now, according to the question we have n=100n = 100. Also we know that e=−1.6×10−19Ce = - 1.6 \times {10^{ - 19}}C. So the total charge of 100100 electrons is
⇒Q=100×(−1.6×10−19)\Rightarrow Q = 100 \times \left( { - 1.6 \times {{10}^{ - 19}}} \right)
⇒Q=−1.6×10−17C\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
⇒W=ΔU\Rightarrow W = \Delta U
⇒W=U2−U1\Rightarrow W = {U_2} - {U_1} …….(2)
Now, as the potential energy is given as
⇒U=qV\Rightarrow U = qV
So we have
⇒U1=QV1\Rightarrow {U_1} = Q{V_1} , and
⇒U2=QV2\Rightarrow {U_2} = Q{V_2}
So from (2) we get
⇒W=QV2−QV1\Rightarrow W = Q{V_2} - Q{V_1}
⇒W=Q(V2−V1)\Rightarrow W = Q\left( {{V_2} - {V_1}} \right) ……..(3)
According to the question, we have
⇒V1=10V\Rightarrow {V_1} = 10{\text{V}}
⇒V2=−4V\Rightarrow {V_2} = - 4{\text{V}}
Also from (1) we have
⇒Q=−1.6×10−17C\Rightarrow Q = - 1.6 \times {10^{ - 17}}C
Substituting these in (3) we get
⇒W=−1.6×10−17(−4−10)\Rightarrow W = - 1.6 \times {10^{ - 17}}\left( { - 4 - 10} \right)
⇒W=22.4×10−17J\Rightarrow W = 22.4 \times {10^{ - 17}}J
Writing in scientific notation, we finally get
⇒W=2.24×10−16J\Rightarrow W = 2.24 \times {10^{ - 16}}J
Thus the work done comes out to be equal to 2.24×10−16J2.24 \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.