📝 Curl as circulation density (15 MCQs)

📖 From Calculus • 16. Topics in vector Calculus • 15 questions available

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What is Curl as circulation density?

Curl as circulation density:
Curl $\nabla \times \mathbf{F}$ measures local rotation or circulation density; its component in any direction gives the circulation per unit area in that plane.

Example:
For $\mathbf{F} = \langle -y, x, 0 \rangle$ , $\nabla \times \mathbf{F} = \langle 0,0,2 \rangle$ , so circulation density is 2 in the $xy$ -plane.

Reason:
This interpretation makes curl intuitive as a measure of local rotation, connecting microscopic behavior to macroscopic circulation via Stokes' theorem.

Easy

Medium

Hard

📝 All Curl as circulation density MCQs

Q1. A velocity field near a point causes nearby particles to rotate counterclockwise when viewed from above the $xy$ -plane. What does this observation most directly suggest about the $z$ -component of the curl at that point?

A.It is negative because counterclockwise motion always produces negative curl

B.It is positive because counterclockwise circulation corresponds to positive

z

-oriented rotation ✅

C.It must be zero because the particles remain near the same point

D.Its sign cannot be determined without knowing the particle speed

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: For a positively oriented loop in the $xy$ -plane, counterclockwise circulation corresponds to positive orientation about the $z$ -axis. The $z$ -component of curl measures the local tendency of the field to circulate around such small loops. Therefore, the observed counterclockwise tendency indicates a positive $z$ -component.

Q2. Two vector fields have the same value of $\nabla\times\mathbf F$ at a point, but one field has a much larger magnitude of velocity there. Which conclusion is most justified?

A.The field with larger velocity must have larger curl

B.The field with smaller velocity must have larger circulation

C.Equal curl means their local circulation density is the same, although their speeds may differ ✅

D.Equal curl means the vector fields are identical near the point

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Curl is not simply the magnitude of the vector field. It describes local rotational tendency or circulation per unit oriented area in the limiting sense. Two fields can therefore have very different velocities while possessing the same curl at a point and consequently the same infinitesimal circulation density.

Q3. A small circular loop is centered at a point where $(\nabla\times\mathbf F)\cdot\mathbf n=5$ . If the loop's area is reduced while its shape remains similar and its normal direction stays fixed, what happens to the circulation approximately?

A.It remains exactly

5

B.It approaches

5

as the area approaches zero

C.It becomes approximately

5

times the loop area ✅

D.It becomes approximately

5

divided by the loop area

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: For a sufficiently small loop, circulation is approximately the normal component of curl multiplied by the enclosed area. Thus, if the curl component is $5$ , the circulation behaves approximately like $5A$ . Reducing the loop area therefore reduces the circulation proportionally, while the circulation density remains near $5$ .

Q4. Suppose a vector field has $\nabla\times\mathbf F=(0,0,-4)$ at a point. A tiny loop is traversed counterclockwise when viewed from above. What is the expected sign of the circulation around the loop?

A.Positive

B.Negative ✅

C.Zero

D.It depends only on the perimeter, not the orientation

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: A counterclockwise traversal viewed from above corresponds to an upward normal, or positive $z$ -direction. Since the curl has $z$ -component $-4$ , its dot product with that normal is negative. Therefore, for a sufficiently small positively oriented loop, the circulation must be negative.

Q5. A rectangular loop in the $xy$ -plane has area $0.02$ square units. Near its center, the $z$ -component of curl is approximately $7$ . If the curl varies negligibly across the loop, what is the best estimate of the circulation for counterclockwise traversal viewed from above?

A.-0.14 ✅

B.-0.35

C.-7.02

D.-140

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The local circulation density is represented by the normal component of curl. For a small planar loop with negligible variation, circulation is approximately $(\nabla\times\mathbf F)\cdot\mathbf n$ times area. Here that gives $7(0.02)=0.14$ , and the counterclockwise orientation agrees with the positive $z$ -normal.

Q6. A field is sampled around two tiny loops of equal area. Loop A is traversed counterclockwise from above and has circulation $0.06$ . Loop B is traversed clockwise from above and has circulation $0.10$ . Which comparison of the corresponding $z$ -components of curl is most reasonable, assuming both loops are sufficiently small?

A.Curl at A is positive and curl at B is positive

B.Curl at A is negative and curl at B is positive

C.Curl at A is positive, while curl at B is negative ✅

D.Both curl components must be zero

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Orientation reverses the sign of circulation relative to a fixed normal. Loop A uses an upward normal, so its positive circulation indicates positive $z$ -curl. Loop B's clockwise traversal corresponds to a downward normal; positive circulation then indicates the curl dotted with the downward normal is positive, meaning the $z$ -component itself is negative.

Q7. A student computes the circulation around a small loop and divides by the loop's perimeter to estimate the curl. What is the main conceptual error?

A.Curl is obtained from circulation divided by enclosed area in the small-loop limit, not perimeter ✅

B.Curl is always equal to circulation multiplied by perimeter

C.Curl cannot be related to circulation in any way

D.Perimeter and area are interchangeable for sufficiently small loops

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Curl represents circulation density with respect to oriented area, not boundary length. For a sufficiently small loop, the relevant approximation is circulation divided by the enclosed oriented area, with the limit taken as the loop shrinks. Dividing by perimeter produces a quantity with different geometric meaning and units.

Q8. A graph of circulation versus enclosed area for similarly shaped tiny loops centered at one point is approximately a straight line through the origin with slope $3$ . What does the slope represent when all loops have the same upward normal?

A.The magnitude of the field

B.The perimeter of the smallest loop

C.The

z

-component of curl at the point ✅

D.The divergence of the field

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: For small loops with the same orientation, circulation is approximately the normal component of curl times area. Therefore, a graph of circulation against area has slope equal to $(\nabla\times\mathbf F)\cdot\mathbf n$ . With an upward normal, this is precisely the $z$ -component of curl, so the slope is $3$ .

Q9. A graph shows that circulation around progressively smaller loops approaches zero, but the ratio of circulation to area approaches $-2$ . Which interpretation is best?

A.Curl must be zero because circulation approaches zero

B.The field has approximately

-2

units of circulation density in the chosen normal direction ✅

C.The field magnitude approaches

-2

D.The circulation becomes independent of loop area

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: It is expected that circulation itself approaches zero as the loop area shrinks. The meaningful local quantity is the ratio of circulation to oriented area. If that ratio approaches $-2$ , then the normal component of curl approaches $-2$ . Thus zero limiting circulation does not imply zero curl.

Q10. Consider a small loop whose plane is tilted relative to a vector $\nabla\times\mathbf F$ . The loop is moved and shrunk around a point while maintaining the same orientation. Which factor controls the leading approximation to its circulation?

A.Only the perimeter of the loop

B.Only the magnitude of the curl, regardless of orientation

C.The dot product of curl with the loop's oriented normal, multiplied by its area ✅

D.The divergence multiplied by the loop's perimeter

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: For a small oriented surface, circulation is governed by the component of curl normal to that surface. The dot product $(\nabla\times\mathbf F)\cdot\mathbf n$ extracts that component, and multiplying by area gives the leading approximation. A tangential component of curl contributes little or nothing to this particular oriented circulation.

Q11. A student walks around a tiny square loop and records a positive circulation. After reversing the walking direction without changing the square or the field, what should happen to the measured circulation?

A.Its sign reverses while its magnitude remains the same ✅

B.Its magnitude doubles

C.Its sign and magnitude both remain unchanged

D.It becomes zero because the path is still closed

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Reversing the traversal direction reverses the orientation of the boundary. Every line-element contribution changes sign, so the total circulation also changes sign. For the same geometric loop and unchanged field, the magnitude remains the same under reversal, assuming the path is traversed in the same manner and no other conditions change.

Q12. Two students estimate local curl using tiny loops. Student A uses a circle with upward normal; Student B uses the same circle but traverses it clockwise as viewed from above. If both obtain positive circulation, what must be true about their reported curl components relative to their chosen normals?

A.Both must report positive normal components ✅

B.A must report positive while B must report negative

C.A must report negative while B must report positive

D.Both must report zero

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Student A's counterclockwise traversal corresponds to an upward normal, so positive circulation gives a positive normal curl component. Student B's clockwise traversal corresponds to a downward normal. Positive circulation relative to that downward normal also means the curl has a positive component along the chosen normal, even though its $z$ -component is negative.

Q13. A vector field produces nearly zero circulation around every tiny loop in the horizontal plane at a point, but substantial circulation around tiny vertical loops. What does this pattern most strongly indicate?

A.The curl is necessarily zero

B.The curl is likely directed mainly horizontally rather than vertically ✅

C.The divergence must be nonzero

D.The field must be constant everywhere

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Horizontal loops test the curl component along the vertical normal, while vertical loops test components lying in horizontal directions. If horizontal loops show negligible circulation but vertical loops show substantial circulation, the normal component of curl is likely small vertically but significant horizontally. This illustrates why circulation depends strongly on loop orientation.

Q14. A model predicts that a tiny loop centered at a point has circulation approximately $4A$ , where $A$ is its area, for counterclockwise traversal from above. Another model predicts approximately $-4A$ . Experimental measurements consistently agree with the first model. Which conclusion is strongest?

A.The field itself must have magnitude

4

B.The

z

-component of curl is approximately

4

at the point ✅

C.The divergence is approximately

4

D.The circulation around every finite loop must equal

4

times its area exactly

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: For sufficiently small counterclockwise loops viewed from above, the upward normal is selected. The relationship between circulation and area is controlled by the normal component of curl. Therefore, experimental agreement with $4A$ indicates that the local $z$ -component of curl is approximately $4$ , without implying anything directly about field magnitude or divergence.

Q15. Suppose a smooth vector field has a nonzero curl at a point, and a family of similar loops centered there is uniformly scaled by a factor $r$ . As $r$ becomes very small, which statement best describes the leading behavior of circulation?

A.It is proportional to

r

B.It is proportional to

r^2

✅

C.It is proportional to

1/r

D.It remains constant for all

r

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Scaling every linear dimension of a planar loop by $r$ scales its enclosed area by $r^2$ . Since local circulation is approximately the normal component of curl multiplied by area, the leading circulation therefore scales as $r^2$ . This result distinguishes circulation from quantities based directly on path length, which would scale linearly with $r$ .

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