Question

Two conducting wires $X$ and $Y$ of the same diameter but different materials are joined in series across a battery. If the number density of electrons in $X$ is twice than in $Y$, find the ration of the drift velocity of electrons in the two wires.

Answer 1

The free electrons present in the conductor undergo collision and acceleration when an external electrical field is applied to the conductor. The electrons acquire the drift velocity due to this motion. They oppose the external electric field by moving in the opposite direction when the electric field applied
Formula used:
$V_{d}=\dfrac{I}{NeA}$

Complete step-by-step solution:
Let us assume that it takes $\Delta t$ time for the electrons with drift speed $V_{d}$ to cover $\Delta x$ distance over the wires. Or,$V_{d}=\dfrac{\Delta x}{\Delta t}$
Let $\Delta Q$ be the no of charges flowing through the conductor during $\Delta t$ time, then $\Delta Q=NeV$, where $N$ is the number of electrons flowing per unit volume and $V$ is the volume of the conductor.
Then, we can also write the volume of the conductor as $V=A\Delta x$. where $A$ is the area of the cross-section of the wire.
Then, we get, $\Delta Q=NeA\Delta x$
Or,$\Delta Q=NeAV_{d}\Delta t$
Then, we $\dfrac{\Delta Q}{\Delta t}=I=NeAV_{d}$ where $I$ is the current flowing per unit time.
Or, $V_{d}=\dfrac{I}{NeA}$
Here, since it is given that the radii of wire $X$ and$Y$ are the same, then their $A$ is also a constant. Since the two wires are connected in series, the same $I$ flows through both of them, hence we can say that $I$ is also a constant. And since the charge of the electron$e$ is also the same that will also be a constant.
Then we get $V_{d}\propto \dfrac{1}{N}$
Let the drift velocity of wire $X$ and$Y$ be $V_{x}$ and $V_{y}$ respectively. Then let their electron density be $N_{x}$ and$N_{y}$ respectively. Given that, $N_{x}=2N_{y}$
Or, $\dfrac{V_{x} }{V_{y}}=\dfrac{N_{y}}{N_{x}}=\dfrac{N_{y}}{2N_{y}}=\dfrac{1}{2}$
Thus the ratio between $\dfrac{V_{x} }{V_{y}}=\dfrac{1}{2}$.

Note: The drift speed is generally given as $V_{d}=\dfrac{-eEt}{m}$ where, the negative sign indicates the opposite direction of the flow of electrons, $e$ is the charge on the electron, $E$ is the electric field applied for $t$ time and $m$ is the mass of one electron.