Are the paths of free electrons straight lines between successive collisions (with the positive ions of metal wire) in the-
(i) Absence of electric field (ii) Presence of electric field

Options

A. `sqrt((mv_(0)^(2))/(12k))`
B. `sqrt((mv_(0)^(2))/(5k))`
C. `sqrt((mv_(0)^(2))/(10k))`
D. None of these

Explanation

To solve the question regarding the paths of free electrons in a metal wire under different conditions, we will analyze both scenarios: the absence of an electric field and the presence of an electric field.

### Step-by-Step Solution:

**Step 1: Analyze the absence of an electric field.**
- In the absence of an electric field, free electrons in a metal wire move randomly due to thermal energy. However, they experience collisions with positive ions (the lattice structure of the metal).
- During their motion, the electrons travel in straight lines between these collisions. The random thermal motion does not influence the straight-line path between collisions.

**Conclusion for Step 1:**
- In the absence of an electric field, the paths of free electrons are straight lines between successive collisions.

**Step 2: Analyze the presence of an electric field.**
- When an electric field is applied to the metal wire, it exerts a force on the free electrons, causing them to accelerate in the direction opposite to the electric field (since electrons are negatively charged).
- Although the electrons may initially move in a straight line due to the electric field, they will still collide with the positive ions of the metal lattice. These collisions will cause the electrons to scatter and change direction.
- As a result, the overall path of the electrons will not be a straight line. Instead, it will be influenced by both the electric field and the random nature of the collisions, leading to a more complex trajectory.

**Conclusion for Step 2:**
- In the presence of an electric field, the paths of free electrons are not straight lines; they are twisted or curved due to the influence of the electric field and collisions.

### Final Conclusion:
- (i) In the absence of an electric field, the paths of free electrons are straight lines between successive collisions.
- (ii) In the presence of an electric field, the paths of free electrons are not straight lines; they are twisted or curved.