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Chapter 27: Problem 29

Find the magnetic-field strength in a region where the magneticenergy density is \(7.8 \mathrm{J} / \mathrm{cm}^{3} .\)

Short Answer

Expert verified
To find the exact numerical value of the magnetic field strength, substitute the given and known quantities into the rearranged magnetic energy density equation and compute the result.

Step by step solution

01

Convert units of magnetic energy density

First, we need to convert the given magnetic energy density from \( J/cm^3 \) to \( J/m^3 \). There are \( 10^6 cm^3 \) in \( 1 m^3 \), so the conversion is \( u = 7.8 J/cm^3 * 10^6 = 7.8 × 10^6 J/m^3 \).
02

Rearrange the magnetic energy density formula

Next, we rearrange the formula for the magnetic energy density to solve for \( B \). This gives \( B = \sqrt{2u\mu_0} \).
03

Substitute and Solve

Substitute the values \( u = 7.8 × 10^6 J/m^3 \) and \( \mu_0 = 4\pi \times 10^{-7} Tm/A \) into the equation to find \( B \). This gives \( B = \sqrt{2 * 7.8 × 10^6 * 4\pi \times 10^{-7}} \). Upon calculating this expression, we find the magnetic field strength \( B \).

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Most popular questions from this chapter

Chapter 26 claimed that a static magnetic field cannot change the energy of a charged particle. Is this true of a changing magnetic field? Discuss.

List some similarities and differences between inductors and capacitors.

A single-turn loop of radius \(R\) carries current \(I\). How does the magnetic- energy density at the loop center compare with that of a long solenoid of the same radius, carrying the same current, and consisting of \(n\) turns per unit length?

The current in a series \(R L\) circuit increases to \(20 \%\) of its final value in \(3.1 \mu \mathrm{s}\). If \(L=1.8 \mathrm{mH},\) what's the resistance?

An electric field and a magnetic field have the same energy density. Find an expression for the ratio \(E / B\) and evaluate this ratio numerically. What are its units? Is your answer close to any of the fundamental constants listed inside the front cover?
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