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Chapter 16: Problem 44

Two "central" fields Find a field \(\mathbf{F}=M(x, y) \mathbf{i}+N(x, y) \mathbf{j}\) in the \(x y\) -plane with the property that at each point \((x, y) \neq(0,0), \mathbf{F}\) points toward the origin and \(|\mathbf{F}|\) is (a) the distance from \((x, y)\) to the origin, (b) inversely proportional to the distance from \((x, y)\) to the origin. (The field is undefined at \((0,0)\) .)

Short Answer

Expert verified
(a) \( \mathbf{F} = -(x\mathbf{i} + y\mathbf{j}) \); (b) \( \mathbf{F} = -\frac{c}{x^2 + y^2}(x\mathbf{i} + y\mathbf{j}) \).

Step by step solution

01

Understanding the Problem

We need to find a vector field \( \mathbf{F} = M(x, y) \mathbf{i} + N(x, y) \mathbf{j} \) such that it points towards the origin at each point \((x, y) eq (0,0)\). This means that the field's direction should always be towards \((0,0)\), which implies that the field vector at each point should be a negative multiple of the position vector \( \mathbf{r} = x\mathbf{i} + y\mathbf{j} \).
02

Express \( \mathbf{F} \) in Terms of \( \mathbf{r} \)

Given that \( \mathbf{F} \) should point towards the origin, we can express it as \( \mathbf{F} = -k(x, y) (x\mathbf{i} + y\mathbf{j}) \), where \( k(x, y) \) is a function that determines the magnitude of \( \mathbf{F} \) according to the specific conditions given in part (a) or (b).
03

Solve for Part (a)

For part (a), \( |\mathbf{F}| \) is the distance from \((x, y)\) to the origin, which is \( \sqrt{x^2 + y^2} \). Thus:\[|\mathbf{F}| = k(x, y) \sqrt{x^2 + y^2} = \sqrt{x^2 + y^2}\]This implies \( k(x, y) = 1 \). Therefore:\[\mathbf{F} = -(x\mathbf{i} + y\mathbf{j})\]
04

Solve for Part (b)

For part (b), \( |\mathbf{F}| \) is inversely proportional to the distance from \((x, y)\) to the origin. This means:\[|\mathbf{F}| = \frac{c}{\sqrt{x^2 + y^2}}\]where \( c \) is a constant. Thus:\[k(x, y) \sqrt{x^2 + y^2} = \frac{c}{\sqrt{x^2 + y^2}}\]Solving for \( k(x, y) \), we have \( k(x, y) = \frac{c}{(x^2 + y^2)} \). Therefore:\[\mathbf{F} = -\frac{c}{x^2 + y^2}(x\mathbf{i} + y\mathbf{j})\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Central Fields
Central fields are fascinating aspects of vector fields. They are specially designed fields where the direction vectors always point towards a central point, often the origin in a coordinate system. In simpler terms, no matter where you are in the plane, the vector field always "pulls" towards that central point. This unique characteristic makes central fields very important in physics, especially in problems involving forces like gravity or electric fields.
To better understand the core idea, consider the following features:
  • Central fields are radially symmetric, meaning their influence spreads evenly from a central point outward in all directions.
  • The strength and direction of the vectors in these fields depend on both their distance and the angle relative to the origin.
Central fields can sometimes represent physical phenomena like gravitational attraction, where every particle moves directed towards a mass at the origin. This concept is crucial for understanding how larger systems function and interact with central forces.
Inverse Proportionality
The idea of inverse proportionality is an essential concept in mathematics and physics, especially when dealing with vector fields. Simply put, when two quantities are inversely proportional, as one quantity increases, the other decreases. The precision of this relationship can be captured in mathematical form: \[ y \propto \frac{1}{x} \]This formula indicates that "y" is inversely proportional to "x".
In the context of the vector field problem, the magnitude of the field vectors is designed to decrease as they move away from the origin. This specific form of inverse proportionality is often observed in gravitational and electrostatic fields where the force experienced by a particle decreases with an increase in distance from the source.
Key aspects include:
  • The field strength will decrease as the distance from the origin increases.
  • This behavior ensures that objects experience less force the further they are from the center.
Inverse proportionality ties in with the field's nature of pulling objects toward a center, balancing the force's reach across different distances.
Position Vectors
Position vectors play a pivotal role in describing locations within a vector field. A position vector essentially indicates the position of a point in space relative to the origin of the coordinate system. For example, the position vector \( \mathbf{r} = x\mathbf{i} + y\mathbf{j} \) precisely describes any point \((x,y)\) in a two-dimensional plane.
The significance of position vectors in our context involves their use as a reference for determining the direction of the vector field. Given that the field should point toward the origin, position vectors help us establish the necessary direction by providing a clear path "backward" along their lines:
  • Position vectors start at the origin and point outward towards a specified point.
  • In our vector field problem, the field vectors are oriented in the opposite direction of these position vectors. This ensures that the entire field consistently points back to the origin.
Understanding position vectors is crucial for visualizing and calculating how a vector field operates throughout a given space. Mastery of this concept enhances one's ability to engage with complex systems modeled by vector fields efficiently.

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Most popular questions from this chapter

Find the outward flux of the field \(\mathbf{F}=x z \mathbf{i}+y z \mathbf{j}+\mathbf{k}\) across the surface of the upper cap cut from the solid sphere \(x^{2}+y^{2}+z^{2} \leq 25\) by the plane \(z=3.\)

Find the outward flux of the field \(\mathbf{F}=2 x y \mathbf{i}+2 y z \mathbf{j}+2 x z \mathbf{k}\) across the surface of the cube cut from the first octant by the planes \(x=a, y=a, z=a.\)

Integrate \(G(x, y, z)=x+y+z\) over the portion of the plane \(2 x+2 y+z=2\) that lies in the first octant.

Regions with many holes Green's Theorem holds for a region \(R\) with any finite number of holes as long as the bounding curves are smooth, simple, and closed and we integrate over each component of the boundary in the direction that keeps \(R\) on our immediate left as we go along (see accompanying figure). $$\begin{array}{l}{\text { a. Let } f(x, y)=\ln \left(x^{2}+y^{2}\right) \text { and let } C \text { be the circle }} \\ {x^{2}+y^{2}=a^{2} . \text { Evaluate the flux integral }}\end{array}$$ $$\oint_{C} \nabla f \cdot \mathbf{n} d s$$ b. Let \(K\) be an arbitrary smooth, simple closed curve in the plane that does not pass through \((0,0) .\) Use Green's Theorem to show that $$\oint_{K} \nabla f \cdot \mathbf{n} d s$$ has two possible values, depending on whether \((0,0)\) lies inside \(K\) or outside \(K .\)

In Exercises \(43-46,\) use a CAS and Green's Theorem to find the counterclockwise circulation of the field \(F\) around the simple closed curve C. Perform the following CAS steps. $$\begin{array}{l}{\text { a. Plot } C \text { in the } x y \text { -plane. }} \\\ {\text { b. Determine the integrand }(\partial N / \partial x)-(\partial M / \partial y) \text { for the tangen- }} \\ {\text { tial form of Green's Theorem. }} \\ {\text { c. Determine the (double integral) limits of integration from }} \\ {\text { your plot in part (a) and evaluate the curl integral for the }} \\ {\text { circulation. }}\end{array}$$ $$\mathbf{F}=(2 x-y) \mathbf{i}+(x+3 y) \mathbf{j}, \quad C : \text { The ellipse } x^{2}+4 y^{2}=4$$
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