Step 1: Meaning of relaxation time.
Electrons in a metal are in constant thermal motion and keep colliding with the vibrating lattice ions. The mean free time between one such collision and the next is called the relaxation time \(\tau\). It measures how long, on average, the field can act on an electron before a collision resets its motion.
Step 2: Set up the equation of motion.
With field \(E\) applied, each electron feels force \(-eE\) (magnitude \(eE\)). Newton's law gives acceleration magnitude \(a = \dfrac{eE}{m}\).
Step 3: Average over many electrons.
Let the velocity of the \(i\)-th electron just after its last collision be \(u_i\); these are random so \(\langle u_i\rangle = 0\). After time \(t_i\) since that collision its velocity is \(v_i = u_i + a t_i\). Averaging over all electrons, the mean of \(u_i\) vanishes and the mean of \(t_i\) is \(\tau\).
Step 4: Obtain drift velocity.
\(v_d = \langle v_i\rangle = \langle u_i\rangle + a\langle t_i\rangle = 0 + a\tau = \dfrac{eE}{m}\tau.\)
Step 5: Final relation.
\[\boxed{v_d = \frac{eE\tau}{m}}\]
Thus a longer relaxation time or a stronger field produces a larger drift velocity, the electrons drifting against the field direction.