Question

A copper rod of cross-sectional area $A$ carries a uniform current $I$ through if. At temperature $T$, if the volume charge density of the rod is$\rho$, how long will the charges take to travel a distance $d$?
A. $\dfrac{{2\rho dA}}{{IT}}$
B. $\dfrac{{2\rho dA}}{I}$
C. $\dfrac{{\rho dA}}{I}$
D. $\dfrac{{\rho dA}}{{IT}}$

Answer 1

This question is solved directly using the formula of charge density. On expanding the singular physical quantities in it, we will get time’s variable as well. Upon rearranging, we find the formula for time and the answer will be given by simple substitution.

Formula Used: Charge density: $\rho = \dfrac{q}{v} = \dfrac{{It}}{{Ad}}$
Where $\rho$ is the charge density on the rod and is expressed in Coulombs per Volt $(C/V)$, $q$ is the charge on the rod and is expressed in Coulombs $(C)$, $v$ is the volume of the rod and is expressed in meter cube $({m^3})$, $I$ is the current in the rod and is expressed in Amperes $(A)$, $t$ is the time period for which the charge $q$ flows in the rod and is expressed in seconds $(s)$, $A$ is the cross-sectional area of the rod and is expressed in meter square $({m^2})$ and $d$ is the distance of the wire in which current flows and is expressed in meters $(m)$.

Complete step by step answer:
We know that the charge density of the rod is $\rho = \dfrac{q}{v}$.
But $q = It$ and $v = Ad$.
Substituting these values in the formula of charge density we get,
$\rho = \dfrac{q}{v} = \dfrac{{It}}{{Ad}}$
Upon rearrangement we will get the following formula for time period $t$.
$\rho = \dfrac{{It}}{{Ad}} \\\ \Rightarrow t = \dfrac{{\rho AD}}{I} \\\$
Therefore, charge takes $\dfrac{{\rho AD}}{I}$ seconds to travel a distance $d$ through a copper rod of area $A$.$$
In conclusion, the correct option is C.

Note: Temperature is not needed to derive us to the answer. It must not be confused with the variable of time. Both are in different cases and one is needed in the derivation while the other is not.