In a metallic conductor, under the effect of applied electric field, the free electrons of the conductor

We begin by considering the motion of free electrons in a metallic conductor under an applied electric field. When an electric field $$\vec{E}$$ is applied across the conductor, it points from higher potential to lower potential, and since electrons carry negative charge, the electric force on them is $$\vec{F} = -e\vec{E}$$. Because this force is opposite to the field direction, electrons experience a force directed from lower potential to higher potential and therefore drift in that direction (from the negative terminal toward the positive terminal), which rules out Option 1.

Next, we examine the nature of the electrons’ paths. Free electrons do not travel in straight lines for several reasons. First, at any non-zero temperature they have random thermal velocities on the order of $$10^5$$ m/s in random directions, which is much greater than the typical drift velocity of $$10^{-4}$$ m/s. Second, they undergo frequent collisions with vibrating lattice ions, impurities, and defects, giving a mean free path on the order of nanometers. Third, between collisions an electron is accelerated by the electric field while retaining a random residual velocity from its previous collision, resulting in a curved (parabolic) trajectory similar to projectile motion.
After each collision, the velocity is randomized and then reaccelerated by the field until the next collision, and the net effect of these curved segments is a slow drift from lower potential to higher potential.

Furthermore, Option 2 is incorrect because it asserts a uniform velocity even though electrons are constantly accelerated between collisions and then scattered randomly. Option 3 is also wrong since the random thermal motion and collisions produce zigzag and curved paths rather than straight lines.

Therefore, free electrons move in curved paths from lower potential to higher potential. Between collisions, the combination of random thermal velocity and the constant electric force creates parabolic trajectories, and the overall drift is toward higher potential. The correct answer is Option 4: Move in the curved paths from lower potential to higher potential.