Question

Is the vector field shown in the figure conservative?

Explain.

Short answer

The vector field is not a conservative

Step-by-step solution

Line integral for a vector field

For a conservative vector field (F), the line integral around any closed path\(C\)is zero. This is mathematically expressed as follows,

\(\int_C {\bf{F}} d{\bf{r}} = 0\)

Check whether the vector field shown in the figure is conservative or not

Consider curve \(C\) is a circle with center at origin and has counterclockwise orientation. The vector field along the curve \(C\) is shown in Figure 1.

In Figure 1, the field is following the same motion of curve\(C\). Hence, the value of integral around curve\(C\)must be positive but not zero. According to definition of conservative, the given vector field is not a conservative.

Thus, the vector field is not a conservative.