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Chapter 28: Problem 55

If you want to construct an electromagnet by running a current of 3.0 A through a solenoid with 500 windings and length \(3.5 \mathrm{~cm}\) and you want the magnetic field inside the solenoid to have magnitude \(B=2.96 \mathrm{~T}\), you can insert a ferrite core into the solenoid. What value of the relative magnetic permeability should this ferrite core have in order to make this work?

Short Answer

Expert verified
Answer: The relative magnetic permeability of the ferrite core needed is approximately 699.43.

Step by step solution

01

Known values and constants

We are given the following information: - Current, \(I = 3.0 \mathrm{~A}\) - Number of windings, \(N = 500\) - Solenoid length, \(l = 3.5 \mathrm{~cm} = 0.035 \mathrm{~m}\) - Magnetic field, \(B = 2.96 \mathrm{~T}\) - Permeability of free space, \(\mu_0 = 4\pi \times 10^{-7} \mathrm{~Tm/A}\)
02

Calculate the number of turns per unit length

We can calculate the number of turns per unit length (\(n\)) by dividing the total number of windings by the length of the solenoid: \(n = \frac{N}{l} = \frac{500}{0.035 \mathrm{~m}} = 14285.71 \mathrm{~turns/m}\)
03

Rearrange the formula for the magnetic field

We rearrange the formula for the magnetic field inside a solenoid with a core to solve for \(\mu_r\): \(\mu_r = \frac{B}{\mu_0 n I}\)
04

Plug in the values and solve for the relative magnetic permeability

Now we can plug in the values and solve for \(\mu_r\): \(\mu_r = \frac{2.96 \mathrm{~T}}{(4\pi \times 10^{-7} \mathrm{~Tm/A})(14285.71 \mathrm{~turns/m})(3.0 \mathrm{~A})} = 699.43\) The relative magnetic permeability of the ferrite core should be approximately 699.43 to create the desired magnetic field.

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Most popular questions from this chapter

Two long, straight parallel wires are separated by a distance of \(20.0 \mathrm{~cm}\). Each wire carries a current of \(10.0 \mathrm{~A}\) in the same direction. What is the magnitude of the resulting magnetic field at a point that is \(12.0 \mathrm{~cm}\) from each wire?

Suppose that the magnetic field of the Earth were due to a single current moving in a circle of radius \(2.00 \cdot 10^{3} \mathrm{~km}\) through the Earth's molten core. The strength of the Earth's magnetic field on the surface near a magnetic pole is about \(6.00 \cdot 10^{-5} \mathrm{~T}\). About how large a current would be required to produce such a field?

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