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Chapter 24: Problem 19

$$\text { Solve the problems in related rates.}$$ The magnetic field \(B\) due to a magnet of length \(l\) at a distance \(r\) is given by \(B=\frac{k}{\left[r^{2}+(l / 2)^{2}\right]^{3 / 2}},\) where \(k\) is a constant for a given magnet. Find the expression for the time rate of change of \(B\) in terms of the time rate of change of \(r\)

Short Answer

Expert verified
\( \frac{dB}{dt} = -\frac{3k \cdot r}{[r^{2}+(l/2)^{2}]^{5/2}} \cdot \frac{dr}{dt} \)

Step by step solution

01

Identify the relationship between variables

The magnetic field \(B\) is given by the formula \(B = \frac{k}{\left[r^{2}+(l/2)^{2}\right]^{3/2}}\). This implies \(B\) is a function of \(r\), the distance. To find how \(B\) changes with time, we'll need to use related rates.
02

Apply the chain rule for differentiation

To find \( \frac{dB}{dt} \), we utilize the chain rule. Since \(B\) is a function of \(r\), and \(r\) is a function of time \(t\), we have:\[\frac{dB}{dt} = \frac{dB}{dr} \cdot \frac{dr}{dt}\]Here, \( \frac{dr}{dt} \) is the rate of change of \(r\) with respect to time.
03

Differentiate \(B\) with respect to \(r\)

We need to find \(\frac{dB}{dr}\). Start by differentiating:\[\frac{dB}{dr} = \frac{d}{dr}\left(\frac{k}{\left[r^{2}+(l/2)^{2}\right]^{3/2}}\right)\]Using the chain rule and power rule, we find:\[\frac{dB}{dr} = -\frac{3k \cdot r}{\left[r^{2}+(l/2)^{2}\right]^{5/2}}\]
04

Substitute the derivative in the related rates formula

Substitute \(\frac{dB}{dr}\) into the related rates formula from Step 2:\[\frac{dB}{dt} = -\frac{3k \cdot r}{\left[r^{2}+(l/2)^{2}\right]^{5/2}} \cdot \frac{dr}{dt}\]This expression represents how \(B\) changes with respect to time based on the change in \(r\) over time \(\frac{dr}{dt}\).

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Key Concepts

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

Magnetic Field
A magnetic field is an invisible area around a magnet where magnetic forces act. The strength of this field at a point varies with the position relative to the magnet. In our context, the magnetic field strength, denoted as \( B \), depends on the distance \( r \) from the magnet and the magnet's length \(l\). This dependency is described by the formula \( B = \frac{k}{\left[r^{2}+(l/2)^{2}\right]^{3/2}} \), where \( k \) is a constant characteristic of the magnet.
  • When the distance \( r \) increases, the magnetic field strength \( B \) typically decreases. This is because the denominator of the fraction, representing the distance, increases.
  • The length \( l \) acts as a stabilizing factor, contributing to the overall expression. It ensures that the field doesn't entirely diminish as \( r \) grows.
Understanding this relationship helps us compute how the field changes as the magnet moves or when its surrounding parameters alter.
Differentiation
Differentiation is a mathematical process used to find the rate at which a function changes at any given point. In related rates problems, you're not just looking for singular values but understanding how varying one quantity affects another over time.
In our exercise, differentiation allows us to establish the connection between the magnetic field \( B \) and the distance \( r \). By differentiating \( B \) with respect to \( r \), we can understand how changes in \( r \) affect \( B \). Through this process, we can then relate those changes to time \( t \) using the chain rule. Here's how it's done:
  • First, identify the function you wish to differentiate. In this case, it's \( B \).
  • Apply the power and chain rules to compute the derivative \( \frac{dB}{dr} \).
  • This result helps express how rapid changes in \( r \) over time influence the magnitude of the magnetic field.
Chain Rule
The chain rule is a fundamental concept in calculus used to differentiate composite functions. It helps tie together relationships when one variable indirectly affects others through another function. In our task, \( B \) is affected by \( r \), which in turn depends on time \( t \). The chain rule provides the tool to express this dependency mathematically.
Here's how it works:
  • We know that \( B \) is a function of \( r \): \( B(r) \).
  • Since \( r \) changes with \( t \), we have a secondary function \( r(t) \).
  • Using the chain rule, we express \( \frac{dB}{dt} \) as \( \frac{dB}{dr} \cdot \frac{dr}{dt} \). This formula captures how \( B \) varies with \( t \) by linking it back through these intermediary steps.
This allows us to efficiently find \( \frac{dB}{dt} \) given \( \frac{dr}{dt} \), thus revealing the rate at which the field strength changes over time as the distance varies.
Time Rate of Change
The time rate of change describes how a variable evolves as time progresses. In related rates problems, this is a cornerstone concept because it precisely characterizes how dynamic quantities interact. For our exercise, it signifies how the magnetic field \( B \) changes with respect to time based on the changing distance \( r \).
Consider these key points:
  • The time rate of change of a value \( B \), denoted as \( \frac{dB}{dt} \), tells us how fast \( B \) increases or decreases as time goes by.
  • If \( \frac{dr}{dt} \), the rate at which \( r \) changes over time, is known, we can leverage the chain rule to find \( \frac{dB}{dt} \).
  • Understanding this helps in predicting how a system behaves over time, which is especially useful in practical and experimental settings involving physics and engineering.
Thus, mastering the concept of time rate of change in related rates problems provides clear insights into how dynamic systems develop with shifting conditions.

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Most popular questions from this chapter

$$\text { Solve the problems in related rates.}$$ The tuning frequency \(f\) of an electronic tuner is inversely proportional to the square root of the capacitance \(C\) in the circuit. If \(f=920 \mathrm{kHz}\) for \(C=3.5 \mathrm{pF},\) find how fast \(f\) is changing at this frequency if \(d C / d t=0.3 \mathrm{pF} / \mathrm{s}\)

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