**Hint:** To find the time taken by an electron to drift from one end of the wire to the other, use the formula $I = nAeV_d$, where I is the current, n is the number density of free electrons, A is the cross-sectional area, e is the electric charge, and $V_d$ is the drift velocity.

**Step-by-Step Solution:**

**Given information:**

Number density of free electrons $n = 8.5 \times 10^{28} \text{m}^{-3}\$

Length of the copper wire $l = 3.0 \text{m}$

Area of the cross-section of the wire $A = 2.0 \times 10^{-6} \text{m}^2$

Current carried by the wire $I = 3.0 \text{A}$

Electric charge $e = 1.6 \times 10^{-19} \text{C}$

Use the formula $I = nAeV_d$ to calculate the drift velocity $V_d$:

$V_d = \dfrac{I}{nAe}$

Plug in the given values to calculate $V_d$:

$\ V_d = \dfrac{3.0 \text{A}}{8.5 \times 10^{28} \text{m}^{-3} \times 2.0 \times 10^{-6} \text{m}^2 \times 1.6 \times 10^{-19} \text{C}}\$

Calculate the value of $V_d$ .

Now, to find the time taken by an electron to drift from one end of the wire to the other, use the formula $\ t = \dfrac{l}{V_d}\$, where ${l}$ is the length of the wire and $V_d$ is the drift velocity.

Plug in the values to calculate ${t}$:

$t = \dfrac{3.0 \text{m}}{1.103 \times 10^{-4} \text{m/s}}$

**Answer:**

The time taken by an electron to drift from one end of the wire to the other is approximately $\ 2.7 \times 10^4\$ seconds.

**Note:** This calculation demonstrates the concept of drift velocity, which is the slow average velocity of electrons in a conductor when an electric current flows. The number density of free electrons, cross-sectional area, and electric charge are crucial in determining the drift velocity and subsequently, the time taken for an electron to traverse the wire. This example illustrates the fundamental principles of electrical conduction in conductors.