📝 Sources and sinks in vector fields (14 MCQs)

📖 From Calculus • 16. Topics in vector Calculus • 14 questions available

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What is Sources and sinks in vector fields?

Sources and sinks in vector fields:
Sources ( $\nabla \cdot \mathbf{F} > 0$ ) have outward flow, sinks ( $\nabla \cdot \mathbf{F} < 0$ ) have inward flow, and zero divergence indicates incompressible flow.

Example:
$\mathbf{F} = \langle x,y,0 \rangle$ has $\nabla \cdot \mathbf{F} = 2 > 0$ , a source at origin.

Reason:
Identifying sources/sinks is vital in fluid dynamics, electromagnetism, and heat transfer to understand flow behavior.

Easy

Medium

Hard

📝 All Sources and sinks in vector fields MCQs

Q1. A vector field is given by $\mathbf{F}(x,y)=\langle x,y\rangle$ . At the origin, the field has zero magnitude, while vectors point outward everywhere nearby. What is the most appropriate interpretation of the origin?

A.It is a source because nearby flow moves outward ✅

B.It is a sink because the field vanishes there

C.It is neither because the field is undefined there

D.It is a saddle point because the field changes direction

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Although the vector field is exactly zero at the origin, nearby vectors point away from it in every direction. This indicates that the origin behaves as a source of the flow. A source concerns the surrounding flow pattern and divergence, not merely whether the vector itself is nonzero at the point.

Q2. For the radial field $\mathbf{F}(x,y)=\langle 2x,2y\rangle$ , which observation best distinguishes a source from a sink?

A.Vectors become shorter as distance from the origin increases

B.Vectors point toward the origin and increase in magnitude

C.Vectors point away from the origin and increase in magnitude ✅

D.Vectors are tangent to circles centered at the origin

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: The vectors are $\langle 2x,2y\rangle$ , so they point radially outward from the origin. Their magnitude is $2\sqrt{x^2+y^2}$ , which increases with distance. This combination of outward direction and positive divergence identifies a source rather than a sink or rotational field.

Q3. A fluid model has velocity field $\mathbf{v}=\langle -3x,-3y\rangle$ . A student claims the origin cannot be a sink because the velocity is zero exactly at the origin. Which response is most accurate?

A.The student is correct because a sink requires nonzero velocity

B.The student is incorrect because surrounding vectors converge toward the origin ✅

C.The student is correct because zero velocity always means no divergence

D.The student is incorrect only if the field has nonzero curl

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: A sink is identified by the behavior of the surrounding flow, not by requiring a nonzero velocity at the central point. Here vectors point inward because both components oppose the corresponding coordinates. Thus nearby particles move toward the origin, making it a sink even though the velocity is zero there.

Q4. Consider $\mathbf{F}(x,y)=\langle x,-y\rangle$ . Along the positive $x$ -axis vectors point away from the origin, while along the positive $y$ -axis they point toward the origin. What conclusion is most justified?

A.The origin is definitely a source

B.The origin is definitely a sink

C.The field has source behavior in every direction

D.The origin exhibits neither pure source nor pure sink behavior ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: The flow moves outward along the $x$ -direction but inward along the $y$ -direction. Therefore particles are not uniformly expelled from or drawn into the origin. The origin has saddle-type behavior rather than being a pure source or sink. Directional inspection prevents an incorrect classification based on only one axis.

Q5. A pollutant concentration model uses $\mathbf{F}(x,y)=\langle x^2,y^2\rangle$ . At $(1,1)$ , the field points northeast. A researcher concludes that this point must be a source. Why is that conclusion insufficient?

A.A source must always have zero field magnitude

B.Direction alone at one point does not establish local source behavior ✅

C.A source can occur only when both components are negative

D.The field must have circular streamlines to contain a source

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: A vector pointing northeast at one location does not by itself establish that the point is a source. Source or sink behavior depends on the local pattern of neighboring vectors and, more fundamentally, the divergence. A single vector direction cannot determine whether flow is locally spreading outward.

Q6. For $\mathbf{F}(x,y)=\langle ax,ay\rangle$ , a modeler wants the origin to act as a sink. Which condition on $a$ guarantees this behavior?

a>0

a=0

a<0

✅

D.Any value of

a

produces a sink

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: When $a>0$ , vectors point outward from the origin, producing source behavior. When $a<0$ , vectors point inward because each component opposes its coordinate, so the origin acts as a sink. The case $a=0$ produces a zero field and therefore does not create source or sink behavior.

Q7. A two-dimensional flow field is $\mathbf{F}(x,y)=\langle 4x,-2y\rangle$ . A small circular group of particles centered at the origin becomes elongated horizontally but its area remains approximately unchanged. Which interpretation best fits the model?

A.The origin is a pure source

B.The origin is a pure sink

C.The flow redistributes particles directionally without net local expansion ✅

D.The field must have zero vectors everywhere

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The divergence is $\frac{\partial(4x)}{\partial x}+\frac{\partial(-2y)}{\partial y}=4-2=2$ , so the stated constant-area behavior would actually conflict with the field. If the observation is taken as reliable, the model is inconsistent. Among the choices, directional redistribution describes the observed deformation, but it should prompt model checking rather than a source or sink classification.

Q8. A student computes $\nabla\cdot\mathbf{F}=0$ for a field and concludes that there can be no sources or sinks anywhere. What is the strongest correction?

A.Zero divergence guarantees a source at every point

B.Zero divergence rules out all vector fields

C.Zero divergence indicates no net local expansion, but special singular behavior may require separate analysis ✅

D.Zero divergence means every vector points toward the origin

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Divergence measures net local expansion or compression where the field is sufficiently regular. A zero divergence means there is no net volumetric expansion locally in the regular region, but singular points can require separate treatment. Therefore the conclusion that sources or sinks are impossible everywhere is too broad.

Q9. A diagram of streamlines shows nearly parallel arrows on the left, increasingly separated arrows near the center, and widely separated arrows on the right. The arrows all point left to right. Which region most strongly suggests a source-like effect?

A.The left region, because arrows are parallel

B.The center region, because the streamlines are spreading apart ✅

C.The right region, because arrows have the largest spacing

D.No region, because sources require arrows pointing in different directions

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: A source-like region is associated with local spreading of trajectories or flow. The center portion shows streamlines separating as they move downstream, indicating expansion of the flow. The absolute spacing on the right is less important than the local change in spacing, so simply choosing the widest final separation is not the best criterion.

Q10. A drainage system is modeled by $\mathbf{F}(x,y)=\langle -2x,-2y\rangle$ . Water initially near $(3,4)$ moves toward the origin. If the same model is valid closer to the origin, what additional conclusion follows from the divergence?

A.The flow must accelerate outward

B.The region acts as a sink with local compression ✅

C.The flow must have zero divergence

D.The origin must be a source because velocity vanishes there

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The field points toward the origin because both components are negative multiples of the coordinates. Its divergence is $(-2-2)=-4$ , which is negative, indicating local convergence or compression. Thus the origin is associated with sink behavior, and the model predicts a decreasing local flow volume rather than outward expansion.

Q11. Two models describe fluid flow near the origin: Model A is $\mathbf{F}_A=\langle x,y\rangle$ , while Model B is $\mathbf{F}_B=\langle -x,-y\rangle$ . Which comparison is correct?

A.Both are sinks, but Model A is stronger

B.Both are sources because their magnitudes increase with distance

C.Model A is a source and Model B is a sink with equal-strength opposite behavior ✅

D.Neither has source or sink behavior because both vanish at the origin

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Model A points outward and has positive divergence, so it represents a source. Model B points inward and has negative divergence, so it represents a sink. Their divergence magnitudes are equal but have opposite signs. The fact that both fields vanish at the origin does not prevent either from having source or sink behavior.

Q12. A graph of a radial flow shows arrows pointing outward near $r=0$ , but their lengths decrease rapidly as $r$ increases. A student says the entire region must still be a source because all arrows point outward. What is the best critique?

A.Outward direction is irrelevant to source behavior

B.Arrow direction alone is insufficient; the local change in flow density must also be considered ✅

C.A source always requires arrows of increasing length

D.Decreasing arrow length proves the field is a sink

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Outward-pointing vectors suggest source-like direction, but source strength depends on how the flow spreads and how vector magnitude changes spatially. A decreasing vector magnitude can alter the divergence and therefore the local classification. The student has used only directional information and ignored the spatial variation that determines net local expansion.

Q13. A computational model predicts a source at the origin. Its field is $\mathbf{F}=\langle x/(x^2+y^2),y/(x^2+y^2)\rangle$ away from the origin. A student differentiates the components only for $x^2+y^2\neq0$ , obtains zero divergence, and declares the origin irrelevant. What is the key flaw?

A.The divergence calculation is impossible away from the origin

B.The field is rotational, so divergence cannot be used

C.The origin is a singular point, so behavior there cannot be inferred solely from regular-point divergence ✅

D.A zero divergence always means the origin is a sink

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Away from the origin, the divergence of this radial field is zero, but the field is undefined at the origin. The singular point can still represent concentrated source behavior even though ordinary divergence vanishes everywhere in the punctured region. Ignoring the singularity is therefore the central flaw in the student's reasoning.

Q14. A square control region contains two localized effects. One produces outward flow with total strength $6$ , while another produces inward flow with total strength $4$ . If no other effects occur, which qualitative conclusion best describes the region as a whole?

A.It behaves like a net source of strength

2

✅

B.It behaves like a net sink of strength

2

C.It has zero net source or sink behavior because two effects cancel completely

D.It must contain only a source because inward flow cannot be combined with outward flow

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The outward contribution is $6$ , while the inward contribution opposes it with magnitude $4$ . Their net effect is $6-4=2$ outward, so the region has net source behavior. This illustrates why multiple local sources and sinks must be combined algebraically rather than classified independently without considering their relative strengths.

🔗 Related Topics (MCQs)

📝 Vector Fields in Calculus MCQs 📝 Graphical Representations of Vector Fields MCQs 📝 Vector field notation and components MCQs 📝 Inverse square vector fields MCQs 📝 Gradient vector fields MCQs

←Previous Topic:Divergence as flux density📝 Sources and sinks in vector fieldsGauss's law for inverse square fields:Next Topic→