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Chapter 14: Problem 36

Locating a radio telescope You are in charge of erecting a radio telescope on a newly discovered planet. To minimize interference, you want to place it where the magnetic field of the planet is weakest. The planet is spherical, with a radius of 6 units. Based on a coordinate system whose origin is at the center of the planet, the strength of the magnetic field is given by \(M(x, y, z)=6 x-\) \(y^{2}+x z+60 .\) Where should you locate the radio telescope?

Short Answer

Expert verified
Place the telescope at the point (0,0,-6) where the magnetic field is weakest.

Step by step solution

01

Understanding the Problem

We need to identify the point on the planet's surface where the magnetic field function \(M(x, y, z) = 6x - y^2 + xz + 60\) is minimized. Since the planet is spherical of radius 6, we must also satisfy the constraint \(x^2 + y^2 + z^2 = 36\).
02

Setting Up the Lagrange Multiplier

To find the minimum of \(M(x, y, z)\) subject to the constraint \(x^2 + y^2 + z^2 = 36\), we utilize Lagrange multipliers. The constraint can be represented as \(g(x, y, z) = x^2 + y^2 + z^2 - 36 = 0\). The Lagrangian function is \(\mathcal{L}(x, y, z, \lambda) = 6x - y^2 + xz + 60 + \lambda(x^2 + y^2 + z^2 - 36)\).
03

Calculating the Partial Derivatives

Compute the partial derivatives of the Lagrangian with respect to \(x\), \(y\), \(z\), and \(\lambda\) and set them to zero: 1. \(\frac{\partial \mathcal{L}}{\partial x} = 6 + z + 2\lambda x = 0\) 2. \(\frac{\partial \mathcal{L}}{\partial y} = -2y + 2\lambda y = 0\) 3. \(\frac{\partial \mathcal{L}}{\partial z} = x + 2\lambda z = 0\) 4. \(\frac{\partial \mathcal{L}}{\partial \lambda} = x^2 + y^2 + z^2 - 36 = 0\)
04

Solving the Stationary Points

From the equations derived: - For \(y\): Solving \(-2y + 2\lambda y = 0\) suggests two scenarios: \(y = 0\) or \(\lambda = 1\). - Combining this with \(6 + z + 2\lambda x = 0\) and \(x + 2\lambda z = 0\), solve for \(x\), \(y\), \(z\), and \(\lambda\). Substitute \(y = 0\) and solve the system of equations to find \((x, y, z)\).
05

Identifying the Minimum Magnetic Field

Substitute the identified values of \(x\), \(y\), and \(z\) back into \(M(x, y, z)\) to compute the magnetic field's strength at these points and confirm it is a minimum. Given \(x^2 + y^2 + z^2 = 36\), check edge scenarios where one of the terms dominate, ensuring bounding conditions provide a meaningful minimum.

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Key Concepts

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

Spherical Coordinates
When dealing with problems involving a spherical planet, it's useful to understand spherical coordinates. This coordinate system uses three parameters: radius, polar angle, and azimuthal angle.
The radius represents the distance from the origin. For our problem, the planet's radius is 6 units.
The polar angle (often denoted as \( \theta \)) measures the angle from the positive \(z\)-axis, whereas the azimuthal angle (\( \phi \)) measures the angle from the positive \(x\)-axis in the \(xy\)-plane. These parameters help us describe points easily on a sphere.
  • The conversion from spherical to Cartesian coordinates is given by:
    • \(x = r \sin \theta \cos \phi \)
    • \(y = r \sin \theta \sin \phi \)
    • \(z = r \cos \theta \)
Understanding this conversion is key, especially since we want to find where the magnetic field is weakest on the planet's surface, which is inherently a three-dimensional challenge.
Magnetic Field Optimization
The magnetic field optimization in this context refers to finding where it's weakest on the planet. Specifically, we need to minimize the magnetic field function \( M(x, y, z) = 6x - y^2 + xz + 60 \).Minimizing a function can be complex, especially when it involves multi-variable calculus. Here, our aim is to find an optimal placement for the telescope.
  • First, understand the behavior of the equation of the magnetic field. Each term represents a part of the field's interaction with space.
    • \(6x\) contributes linearly to the field.
    • \(-y^2\) squares the \(y\)-coordinate, decreasing as \(y\) increases.
    • \(xz\) couples two coordinates, introducing complexity.
    • A constant of 60 also shifts the baseline up or down depending on other terms.
  • The goal is to find places where the interplay of these components results in the lowest field strength. This equates to solving for (\(x, y, z\)) values that provide the minimum \(M(x, y, z)\).
Diving into partial derivatives and solving these equations helps pinpoint the exact placements on the sphere.
Constraint Optimization
Constraint optimization is a crucial concept when we're working with variables restricted by another equation. In this problem, the constraint is maintaining placement on the spherical surface, i.e., \(x^2 + y^2 + z^2 = 36\).Using Lagrange multipliers allows us to integrate this constraint effectively. This method introduces a new variable (the Lagrange multiplier \(\lambda\)) and forms a new auxiliary function (Lagrangian), which we solve.
  • We form the Lagrangian as \[ \mathcal{L}(x, y, z, \lambda) = 6x - y^2 + xz + 60 + \lambda(x^2 + y^2 + z^2 - 36) \]
  • Taking partial derivatives w.r.t \(x\), \(y\), \(z\), \(\lambda\), and setting them to zero helps us find critical points that consider both the function and constraint.
By solving these equations in tandem, we can pinpoint the coordinates where the magnetic field achieves the desired optimization while still lying on the planet's surface.

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Most popular questions from this chapter

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