Question

An insulating pipe of cross-section area 'A' contains an electrolyte which has two types of ions, their charges being $- e$ and $+ 2e$ . A potential difference applied between the ends of the pipe results in the drifting of the two types of ions, having drift speed $= v( - ve)$ ion and $\dfrac{v}{4}( + ve)$ ion . Both ions have the same number per unit volume $= n$ . The current flowing through the pipe is.
(A) $nev\dfrac{A}{2}$
(B) $nev\dfrac{A}{4}$
(C) $5nev\dfrac{A}{2}$
(D) $4nev\dfrac{A}{2}$

Answer 1

First find the total current in the pipe due to both the given charges then use the formula to find the current using drift velocity, area of cross-section, charge, and number of ions per unit volume. We will get our required current in the pipe.
$I = neA \times {V_d}$
Where, $n$ is the number of ions per unit volume. $A$ is the area of cross-section. $e$ is the charge. ${V_d}$ is the drift velocity.

Complete Step By Step Answer:
We have been given a pipe of cross section ‘A’ inside which there is an electrolyte which has two charges $- e$ and $+ 2e$ their respective drift velocity are, $v( - ve)$ and $\dfrac{v}{4}( + ve)$ .
The total current flowing through the pipe will be due to both the charges $- e$ and $+ 2e$
$\Rightarrow {I_{total}} = {I_{ + 2e}} - {I_{ - e}}$
${I_{ + 2e}}$ ​ is due to the positive ion moving along the field.
${I_{ - e}}$ is due to the negative ion moving opposite to the field.
The equation for the relationship between current and drift velocity is
$I = neA \times {V_d}$
Where, $n$ is the number of ions per unit volume. $A$ is the area of cross-section. $e$ is the charge. ${V_d}$ is the drift velocity.
Since both ions have the same number per unit volume , then the current will be
$\Rightarrow {I_{total}} = n \times (2e) \times A \times \dfrac{v}{4} - \left( {n \times ( - e) \times A \times ( - v)} \right)$
$\Rightarrow {I_{total}} = nev\dfrac{A}{2}$
Hence option A) $nev\dfrac{A}{2}$ is the correct option.

Note:
The average velocity gained by free electrons in a conductor under the effect of an electric field applied across the conductor is known as drift velocity. If an electric field is applied to a solution, the positive ions will migrate inside the solution (following the field), whereas the negative ions will move in the opposite direction.