Question

A bar magnet is often modeled as a magnetic dipole with one end labeled the north pole $N$ and the opposite end labeled the south pole $S .$ The magnetic field for the magnetic dipole is symmetric with respect to rotation about the axis passing lengthwise through the center of the bar. Hence we can study the magnetic field by restricting ourselves to a plane with the bar magnet centered on the $x$ -axis. For a point $P$ that is located a distance $r$ from the origin, where $r$ is much greater than the length of the magnet, the magnetic field lines satisfy the differential equation $$ \frac{d y}{d x}=\frac{3 x y}{2 x^{2}-y^{2}} $$ and the equipotential lines satisfy the equation $$ \frac{d y}{d x}=\frac{y^{2}-2 x^{2}}{3 x y} $$ $$\begin{array}{l}{\text { (a) Show that the two families of curves are perpendicu- }} \\ {\text { lar where they intersect. [Hint: Consider the slopes }} \\ {\text { of the tangent lines of the two curves at a point of }} \\ {\text { intersection. }}\end{array}$$ $$ \begin{array}{l}{\text { (b) Sketch the direction field for equation }(4) \text { for }} \\ {-5 \leq x \leq 5,-5 \leq y \leq 5 . \text { You can use a software }} \\ {\text { package to generate the direction field or use the }} \\ {\text { method of isoclines. The direction field should remind }} \\ {\text { you of the experiment where iron filings are sprinkled }} \end{array} $$$$\begin{array}{l}{\text { on a sheet of paper that is held above a bar magnet. }} \\ {\text { The iron filings correspond to the hash marks. }}\end{array}$$ $$\begin{array}{l}{\text { (c) Use the direction field found in part (b) to help sketch }} \\ {\text { the magnetic field lines that are solutions to }(4) .}\end{array}$$ $$ \begin{array}{l}{\text { (d) Apply the statement of part (a) to the curves in }} \\ {\text { part (c) to sketch the equipotential lines that are }} \\ {\text { solutions to }(5) . \text { The magnetic field lines and the }} \\ {\text { equipotential lines are examples of orthogonal }} \\ {\text { trajectories. (See Problem 32 in Exercises } 2.4 \text { , }} \\ {\text { page } 65 . )^{\dagger}}\end{array} $$

Name: Problem 20, Chapter 1: Fundamentals of Differential Equations
Uploaded: 2019-07-21T17:57:22-08:00
Duration: 5 min 6 s
Channel: Dan Rhoads

   A bar magnet is often modeled as a magnetic dipole with one end labeled the north pole $N$ and the opposite end labeled the south pole $S .$ The magnetic field for the magnetic dipole is symmetric with respect to rotation about the axis passing lengthwise through the center of the bar. Hence we can study the magnetic field by restricting ourselves to a plane with the bar magnet centered on the $x$ -axis. For a point $P$ that is located a distance $r$ from the origin, where $r$ is much greater than the length of the magnet, the magnetic field lines satisfy the differential equation $$ \frac{d y}{d x}=\frac{3 x y}{2 x^{2}-y^{2}} $$ and the equipotential lines satisfy the equation $$ \frac{d y}{d x}=\frac{y^{2}-2 x^{2}}{3 x y} $$
$$\begin{array}{l}{\text { (a) Show that the two families of curves are perpendicu- }} \\ {\text { lar where they intersect. [Hint: Consider the slopes }} \\ {\text { of the tangent lines of the two curves at a point of }} \\ {\text { intersection. }}\end{array}$$
$$
\begin{array}{l}{\text { (b) Sketch the direction field for equation }(4) \text { for }} \\ {-5 \leq x \leq 5,-5 \leq y \leq 5 . \text { You can use a software }} \\ {\text { package to generate the direction field or use the }} \\ {\text { method of isoclines. The direction field should remind }} \\ {\text { you of the experiment where iron filings are sprinkled }} \end{array}
$$$$\begin{array}{l}{\text { on a sheet of paper that is held above a bar magnet. }} \\ {\text { The iron filings correspond to the hash marks. }}\end{array}$$
$$\begin{array}{l}{\text { (c) Use the direction field found in part (b) to help sketch }} \\ {\text { the magnetic field lines that are solutions to }(4) .}\end{array}$$
$$
\begin{array}{l}{\text { (d) Apply the statement of part (a) to the curves in }} \\ {\text { part (c) to sketch the equipotential lines that are }} \\ {\text { solutions to }(5) . \text { The magnetic field lines and the }} \\ {\text { equipotential lines are examples of orthogonal }} \\ {\text { trajectories. (See Problem 32 in Exercises } 2.4 \text { , }} \\ {\text { page } 65 . )^{\dagger}}\end{array}
$$

Fundamentals of Differential Equations

R. Kent Nagle 9th Edition

Chapter 1, Problem 20

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Solved by Expert Dan Rhoads

07/21/2019

Step 1

We are asked to show that these two families of curves are perpendicular where they intersect. To do this, we need to show that the slopes of the tangent lines of the two curves at a point of intersection are negative reciprocals of each other. The slopes of the Show more…

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A bar magnet is often modeled as a magnetic dipole with one end labeled the north pole $N$ and the opposite end labeled the south pole $S .$ The magnetic field for the magnetic dipole is symmetric with respect to rotation about the axis passing lengthwise through the center of the bar. Hence we can study the magnetic field by restricting ourselves to a plane with the bar magnet centered on the $x$ -axis. For a point $P$ that is located a distance $r$ from the origin, where $r$ is much greater than the length of the magnet, the magnetic field lines satisfy the differential equation $$ \frac{d y}{d x}=\frac{3 x y}{2 x^{2}-y^{2}} $$ and the equipotential lines satisfy the equation $$ \frac{d y}{d x}=\frac{y^{2}-2 x^{2}}{3 x y} $$ $$\begin{array}{l}{\text { (a) Show that the two families of curves are perpendicu- }} \\ {\text { lar where they intersect. [Hint: Consider the slopes }} \\ {\text { of the tangent lines of the two curves at a point of }} \\ {\text { intersection. }}\end{array}$$ $$ \begin{array}{l}{\text { (b) Sketch the direction field for equation }(4) \text { for }} \\ {-5 \leq x \leq 5,-5 \leq y \leq 5 . \text { You can use a software }} \\ {\text { package to generate the direction field or use the }} \\ {\text { method of isoclines. The direction field should remind }} \\ {\text { you of the experiment where iron filings are sprinkled }} \end{array} $$$$\begin{array}{l}{\text { on a sheet of paper that is held above a bar magnet. }} \\ {\text { The iron filings correspond to the hash marks. }}\end{array}$$ $$\begin{array}{l}{\text { (c) Use the direction field found in part (b) to help sketch }} \\ {\text { the magnetic field lines that are solutions to }(4) .}\end{array}$$ $$ \begin{array}{l}{\text { (d) Apply the statement of part (a) to the curves in }} \\ {\text { part (c) to sketch the equipotential lines that are }} \\ {\text { solutions to }(5) . \text { The magnetic field lines and the }} \\ {\text { equipotential lines are examples of orthogonal }} \\ {\text { trajectories. (See Problem 32 in Exercises } 2.4 \text { , }} \\ {\text { page } 65 . )^{\dagger}}\end{array} $$

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Key Concepts

Orthogonal Trajectories

Orthogonal trajectories are families of curves that intersect each other at right angles. In the context of magnetic fields, the magnetic field lines and the equipotential lines serve as an example of orthogonal trajectories. Demonstrating their orthogonality involves showing that the product of their slopes at any intersection point is -1, which is a key property used in analyzing and designing field structures.

Equipotential Lines

Equipotential lines are curves along which a scalar potential is constant. In magnetic contexts, they represent locations of equal magnetic potential and are always perpendicular to the magnetic field lines. This orthogonality simplifies the analysis of forces and fields, as it helps in understanding how the field interacts with objects and influences energy distribution.

Direction Fields

Direction fields, or slope fields, are graphical representations of differential equations that display the slope of the solution curve at any given point. This visual tool is essential in understanding the qualitative behavior of solutions to differential equations without necessarily finding an explicit solution. In applications like mapping the magnetic field lines, direction fields help reveal the pattern and structure of the field over a specified region.

Magnetic Dipole

A magnetic dipole is a fundamental model in electromagnetism representing the simplest configuration of a magnetic field. It is characterized by two opposite magnetic poles (north and south) and a magnetic moment that determines the field strength and orientation. The dipole model is important because it captures the essence of more complex magnetic phenomena, and its field decays with distance in a specific pattern that is useful in approximations where the observer is much farther away than the size of the magnet.

Magnetic Field Lines

Magnetic field lines are a visualization tool used to represent the direction and strength of a magnetic field. They are tangent to the magnetic field vector at every point and indicate the path along which a free magnetic pole would move. In the dipole model, these lines exhibit rotational symmetry about the dipole axis, providing insight into the spatial variation of the magnetic influence.

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Transcript

00:01 All right, so we're given these two differential equations, one which represents the magnetic field lines of a magnet and the other, the equipotential field lines.

00:15 And so for the first part, we're supposed to show that these two families of curves, the solution curves, for each of these problems or each of these equations, that there are perpendicular to each other at points where they interstate.

00:34 So first of all, to find just finding where points where they intersect, we could set these two equations equal to each other.

00:44 And by setting the first derivative of these equations, which is what we're given, equal to each other, it's the same thing as the finding the slope of the tangent lines of the solution curves.

01:07 And if we can find the relationship that the slopes of the tangent lines at these points of intersection are negative reciprocal of each other, that is how we could show that they are perpendicular.

01:22 So again, each one of these equations individually would represent the slope of the tangent line at some point.

01:28 Setting them equal to each other is finding locations where the two curves intersect each other.

01:37 And if we can show that these give us a slope with a relationship being negative reciprocal of one another, then we can make the conclusion that they are indeed perpendicular at their points of intersection.

01:52 Okay, after setting these equal to each other, i see that these two terms are the same, and i could cancel them.

01:58 The other two terms are really close, and they're actually just off by a negative sign.

02:03 So i could rewrite this.

02:04 After canceling those terms, i would have a 1 over 2x squared minus y squared.

02:11 And in the top, i'm going to factor out a negative sign.

02:13 And if i do that, and i'm going to change just the order of how i'm writing them, it would give us a positive 2x squared and a minus y squared on the inside.

02:21 That's over 1.

02:23 Now you see that these two terms are the same.

02:25 They cancel.

02:27 And so the relationship here between their slopes is a difference of negative 1.

02:34 Which we could write this as this negative one as one over negative one...

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