The three figures below show three vector fields In each cas

The three figures below show three vector fields. In each case, determine whether the line integral around the circle (oriented counterclockwise) is positive, negative, or zero.

Solution

A) On the upper half of the circle, the vector field makes an angle greater than pi/2 and less than pi with the tangent vector to the circle, so dot product must be negative and for the lower half of the circle, the angle between the vector field and tangent vector is less than pi/2, so the dot product must be positive.

therefore, we can see that the line integral is negative for upper half and positive for lower half. so the total line integral will be zero.

Answer: 0

B) Here, the vector field perpendicular to the tangent vectors at point on the circle. So, the dot product is zero and hence, the line integral is zero. Ans: 0

C) The vector field and the tangent vectors to the circle makes angle greater than pi/2 and less than pi in the first quadrant if we divide the circle into four quadrants. So, the dot product is negative and hence the line integral will be negative.

Now, in rest of the three quadrants the vector and the tangent vectors are making angle less than pi/2 so, the dot product is positive. therefore, the resultant line integral over the complete circle will be positive.

Ans: P