Step 1: Concept Identification:
This query concerns the movement of free electrons within a metallic conductor under two distinct scenarios: the absence and presence of an external electric field. The objective is to characterize the trajectory of an electron between successive collisions with lattice ions.
Step 2: Detailed Analysis:
(i) No Electric Field:
Free electrons in metals exhibit continuous random motion driven by thermal energy. In the interval between two consecutive impacts with the positive metal ions, no net force (excluding negligible gravitational forces) acts upon an electron. As per Newton's first law, an object in motion persists in its state of constant velocity and direction unless acted upon by an external unbalanced force. Consequently, the electron traverses a straight line at a uniform velocity between collisions.
(ii) With Electric Field:
Upon application of an external electric field (\(\vec{E}\)), each electron is subjected to a constant electrostatic force \(\vec{F} = -e\vec{E}\). This uniform force imparts a constant acceleration (\(\vec{a} = \vec{F}/m_e = -e\vec{E}/m_e\)) to the electron, directed opposite to the field. An object possessing an initial velocity and experiencing constant acceleration follows a parabolic path. Therefore, the electron's path between successive collisions becomes curved. The overall motion is a superposition of random thermal velocity and a directional drift induced by the electric field.
Step 3: Conclusion:
In the absence of an electric field, the electron's path is linear. In the presence of an electric field, the path is curved due to the imposed constant electrostatic force.