Question

If \( \vec{F} \) be the force and C is a non-closed arc, then \( \int_C \vec{F} \cdot d\vec{r} \) represents:

• A. Flux
• B. Circulation
• C. Work done
• D. Conservative field

Hint:

Remember the key physical interpretations of vector integrals:
\( \int_C \vec{F} \cdot d\vec{r} \) (line integral of tangential component) = Work (if \(\vec{F}\) is force) or Circulation (if C is closed).
\( \iint_S \vec{F} \cdot d\vec{S} \) (surface integral of normal component) = Flux.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This question asks for the physical interpretation of the line integral of a vector field representing force along a path.

Step 2: Detailed Explanation:
Let's analyze the physical meaning of the given integral and the options.
Work Done: In physics, the work done by a variable force \( \vec{F} \) in moving a particle along a path C is defined as the line integral of the force along that path. The differential work \( dW \) done by the force \( \vec{F} \) over a differential displacement \( d\vec{r} \) is \( dW = \vec{F} \cdot d\vec{r} \). The total work is the sum (integral) of these differential works along the curve C. Thus, \( W = \int_C \vec{F} \cdot d\vec{r} \). This perfectly matches the question.
Flux: The flux of a vector field across a surface (in 3D) or through a curve (in 2D) measures the rate of flow of the field through the surface/curve. The line integral for flux in 2D is \( \int_C \vec{F} \cdot \hat{n} \, ds \), where \( \hat{n} \) is the normal vector to the curve. This is different from the given integral.
Circulation: Circulation is the line integral of a vector field around a closed loop (\( \oint_C \vec{F} \cdot d\vec{r} \)). It measures the tendency of the field to "circulate" around the loop. The question specifies that C is a non-closed arc, so this term is not appropriate.
Conservative Field: A conservative field is a property of the vector field \( \vec{F} \), not a quantity represented by a single integral. A field is conservative if the line integral between any two points is independent of the path taken (which is equivalent to its curl being zero). While the work done by a conservative field has special properties, the integral itself represents work, not the field's property.
Step 3: Final Answer:
The line integral of a force field along a path represents the work done by that force.