Step 1: Understanding the Question:
The question asks for the work done by a magnetic force on a moving charge.
Step 2: Key Formula or Approach:
1. Magnetic force \( \vec{F}_m = q(\vec{V} \times \vec{B}) \).
2. Work done \( W = \int \vec{F} \cdot d\vec{s} \).
Step 3: Detailed Explanation:
The magnetic force \( \vec{F}_m \) is the result of a cross product between \( \vec{V} \) and \( \vec{B} \).
By definition of the cross product, \( \vec{F}_m \) is always perpendicular to both \( \vec{V} \) and \( \vec{B} \).
Since \( \vec{F}_m \perp \vec{V} \), the dot product \( \vec{F}_m \cdot \vec{V} = 0 \).
The displacement \( d\vec{s} \) is in the direction of velocity \( \vec{V} \).
Thus, \( \vec{F}_m \cdot d\vec{s} = F_m \cdot ds \cos(90^\circ) = 0 \).
Therefore, the work done is zero.
Step 4: Final Answer:
The work done by the Lorentz magnetic force is zero.