Question

A small insect crawls in the direction of electron drift along copper wire that carries a current of $2.56A$. It travels with drift speed of the electron in the wire of uniform cross section area $1m{{m}^{2}}$. Number of free electrons for copper = $8\times {{10}^{22}}\,c{{c}^{-1}}$ and resistivity of copper = $1.6\times {{10}^{-8}}\Omega m$. How much time would the insect take to crawl $1.0cm$ if it crawls at the drift speed of the electrons in the wire?
(A). $50s$
(B). $5s$
(C). $500s$

Answer 1

Hint: Given that an insect crawls with the drift velocity of electrons in the copper wire. Drift velocity is the average velocity of electrons flowing in a material. It depends on the current, electron density and the area of the cross section. Convert the units as required, it is recommended to convert units in SI.

Formula used:
${{v}_{d}}=\dfrac{I}{neA}$

Complete step-by-step answer:
Current is the flow of charges per unit time. Conventionally the direction of current is taken opposite to the direction of flow of charges. Drift velocity is the average velocity of charges when they flow in a material due to applied electric field at its ends. The drift velocity is given by-
${{v}_{d}}=\dfrac{I}{neA}$ - (1)
Here, ${{v}_{d}}$ is the drift velocity
$I$ is the current
$n$ is the number of electrons per unit volume or the electron density
$A$ is the area of cross section
Given, $I=2.56A$, $n=8\times {{10}^{22}}\,c{{c}^{-1}}$, $A=1m{{m}^{2}}$
Given values are substituted in the above equation to get,
$\begin{align}
  & {{v}_{d}}=\dfrac{2.56}{8\times {{10}^{22}}\times {{10}^{6}}{{m}^{-3}}\times 1.6\times {{10}^{-19}}\times {{10}^{-6}}{{m}^{2}}} \\
 & \Rightarrow {{v}_{d}}=2\times {{10}^{-4}}m{{s}^{-1}} \\
\end{align}$
Therefore, the drift velocity at which the electrons are moving inside the copper wire is $2\times {{10}^{-4}}m{{s}^{-1}}$ .
The speed of the insect is equal to the speed of the electrons in the copper wire.
We know that,
$v=\dfrac{d}{t}$
Here, $v$ is the velocity
$d$ is the distance travelled
$t$ is the time taken
Given, $d=1.0cm$ substituting values in the above equation, we get,
$\begin{align}
  & v=\dfrac{{{10}^{-2}}}{t} \\
 & \Rightarrow 2\times {{10}^{-4}}m{{s}^{-1}}\times t={{10}^{-2}} \\
 & \therefore t=50s \\
\end{align}$
Therefore, the time taken by the insect to crawl at the drift speed of the electrons is $50s$.

So, the correct answer is “Option A”.

Note: Even though current has both magnitude and direction, it is a scalar quantity. The drift velocity also depends on the temperature and the electric field. Drift velocity does not depend on the resistivity of a material. Resistivity is a constant and depends on the material.