Column - I gives certain physical terms associated with flow of current through a metallic conductor. Column - II gives some mathematical relations involving electrical quantities. Match Column - I and Column - II with appropriate relations.

To match the physical terms associated with the flow of current in a metallic conductor with their corresponding mathematical relations, we derive each term step-by-step:

1. Drift Velocity ($v_d$)
When an electric field $E$ is applied across a metallic conductor, the free electrons experience an electric force:
$F=eE$
The acceleration $a$ experienced by an electron of mass $m$ is given by Newton's second law:
$a=\frac{F}{m}=\frac{eE}{m}$
The drift velocity $v_d$ is the average velocity acquired by the free electrons under the influence of this electric field over the relaxation time $\tau$:
$v_d=a\tau =\frac{eE\tau }{m}$
Thus, (A) matches with (R).

2. Electrical Resistivity ($\rho$)
The electric current $I$ flowing through a conductor of cross-sectional area $A$ is related to the drift velocity $v_d$ by:
$I=neA{v}_d$
Substituting the expression for drift velocity $v_d=\frac{eE\tau }{m}$:
$I=neA\left(\frac{eE\tau }{m}\right)=\frac{n{e}^{2}AE\tau }{m}$
Since the electric field $E=\frac{V}{l}$ (where $V$ is the potential difference and $l$ is the length of the conductor):
$I=\frac{n{e}^{2}AV\tau }{ml}$
Rearranging for resistance $R=\frac{V}{I}$:
$R=\left(\frac{m}{n{e}^2\tau }\right)\frac{l}{A}$
Comparing this with the standard formula for resistance $R=\rho \frac{l}{A}$ yields:
$\rho =\frac{m}{n{e}^2\tau }$
Thus, (B) matches with (S).

3. Relaxation Time ($\tau$)
Using the relation for electrical resistivity $\rho =\frac{m}{n{e}^2\tau }$, we can rearrange the equation to express the relaxation time $\tau$:
$\tau =\frac{m}{n{e}^2\rho }$
Thus, (C) matches with (P).

4. Current Density ($J$)
Current density $J$ is defined as the current flowing per unit cross-sectional area:
$J=\frac{I}{A}$
Substituting the relation $I=neA{v}_d$:
$J=\frac{neA{v}_d}{A}$
Thus, (D) matches with (Q).

Combining all the matches:
(A) ⇒ (R)
(B) ⇒ (S)
(C) ⇒ (P)
(D) ⇒ (Q)
Therefore, the correct matching option is (A)-(R), (B)-(S), (C)-(Q) is incorrect, the correct sequence is (A)-(R), (B)-(S), (C)-(P), (D)-(Q).