A copper rod of cross-sectional area A carries a uniform current I through it. At temperature T, if the volume charge density of the rod is $$\rho$$, how long will the charges take to travel a distance d?

We have a uniform current $$I$$ flowing through a copper rod of cross-sectional area $$A$$. By definition, the current density is the current per unit area, so

$$J \;=\;\frac{I}{A}.$$

For charge carriers moving with an average drift velocity $$v_d$$, the current density can also be written in terms of the volume charge density. The volume charge density $$\rho$$ represents the amount of charge (not merely the number of carriers) present per unit volume. Each tiny volume element therefore carries charge $$\rho$$, and as this charge drifts with speed $$v_d$$, the amount of charge crossing unit area per unit time is

$$J \;=\;\rho\,v_d.$$

Equating the two expressions for $$J$$, we obtain

$$\frac{I}{A}\;=\;\rho\,v_d.$$

Now we solve this equation for the drift velocity $$v_d$$. Dividing both sides by $$\rho$$ gives

$$v_d \;=\;\frac{I}{\rho\,A}.$$

The time $$t$$ required for the charge to travel a distance $$d$$ is simply the distance divided by the drift velocity, because speed equals distance over time. Hence

$$t \;=\;\frac{d}{v_d}.$$

Substituting the expression we just found for $$v_d$$, we get

$$t \;=\;\frac{d}{\dfrac{I}{\rho\,A}} \;=\;d\;\frac{\rho\,A}{I}.$$

Simplifying the numerator and denominator, the final expression for the time is

$$t \;=\;\frac{\rho\,d\,A}{I}.$$

Observe that the temperature $$T$$ does not explicitly enter this relationship, so it does not appear in the final formula.

Hence, the correct answer is Option C.