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Chapter 19: Problem 86

An electromagnet is made by inserting a soft iron core into a solenoid. The solenoid has 1800 turns, radius \(2.0 \mathrm{cm},\) and length $15 \mathrm{cm} .\( When \)2.0 \mathrm{A}$ of current flows through the solenoid, the magnetic field inside the iron core has magnitude 0.42 T. What is the relative permeability \(\overline{K_{\mathrm{B}}}\) of the iron core? (See Section 19.10 for the definition of \(\boldsymbol{K}_{\mathbf{B}} .\) )

Short Answer

Expert verified
Answer: The relative permeability of the iron core is 139.

Step by step solution

01

Write down the formula for the magnetic field inside a solenoid

The formula for the magnetic field (\(B\)) inside a solenoid with \(n\) turns per unit length carrying a current \(I\) is: \(B = μ_0nI\) where \(μ_0\) is the permeability of free space, which is \(4\pi × 10^{-7}\) Tm/A.
02

Identify the information given in the problem and compute the turns per unit length of the solenoid

The information given in the problem is: - Solenoid turns: 1800 - Solenoid radius: 2.0 cm - Solenoid length: 15 cm - Current through the solenoid: 2.0 A - Magnetic field inside the iron core: 0.42 T First, we need to find the number of turns per unit length (\(n\)). We do this by dividing the total number of turns (1800) by the length of the solenoid (15 cm or 0.15 m): \(n = \frac{1800}{0.15 \mathrm{m}} = 12000 \mathrm{m}^{-1}\)
03

Rearrange the formula and use it to find the permeability of the iron core

Now, we need to find the permeability of the iron core (\(μ\)), which is related to the product of the permeability of free space (\(μ_0\)) and the relative permeability of the iron core (\(\overline{K_B}\)): \(μ = μ_0\overline{K_B}\) Using the formula for the magnetic field inside a solenoid, we can get the permeability of the iron core by rearranging the formula as follows: \(μ = \frac{B}{nI}\) Then, we can plug in the values given in the problem: \(μ = \frac{0.42 \mathrm{T}}{12000 \mathrm{m}^{-1} \times 2.0 \mathrm{A}} = 1.75 × 10^{-5} \mathrm{Tm/A}\)
04

Calculate the relative permeability of the iron core

Now that we have the permeability of the iron core (\(μ\)), we can find the relative permeability \(𝑘_B\) by using the relation between \(μ\), \(μ_0\), and \(𝑘_B\): \(\overline{K_B} = \frac{μ}{μ_0}\) Plugging in the values of \(μ\) and \(μ_0\): \(\overline{K_B} = \frac{1.75 × 10^{-5} \mathrm{Tm/A}}{4\pi × 10^{-7} \mathrm{Tm/A}} = 139\)
05

Final Answer

The relative permeability of the iron core is 139.

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