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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

A horizontally oriented coil of wire of radius \(5.00 \mathrm{~cm}\) and carrying a current, \(i\), is being levitated by the south pole of a vertically oriented bar magnet suspended above the center of the coil. If the magnetic field on all parts of the coil makes an angle \(\theta\) of \(45.0^{\circ}\) with the vertical, determine the magnitude and the direction of the current needed to keep the coil floating in midair. The magnitude of the magnetic field is \(B=0.0100 \mathrm{~T}\), the number of turns in the coil is \(N=10.0\), and the total coil mass is \(10.0 \mathrm{~g}\).

A long, straight wire is located along the \(x\) -axis \((y=0\) and \(z=0)\). The wire carries a current of \(7.00 \mathrm{~A}\) in the positive \(x\) -direction. What is the magnitude and the direction of the force on a particle with a charge of 9.00 C located at \((+1.00 \mathrm{~m},+2.00 \mathrm{~m}, 0),\) when it has a velocity of \(3000 . \mathrm{m} / \mathrm{s}\) in each of the following directions? a) the positive \(x\) -direction c) the negative \(z\) -direction b) the positive \(y\) -direction

A long solenoid (diameter of \(6.00 \mathrm{~cm}\) ) is wound with 1000 turns per meter of thin wire through which a current of 0.250 A is maintained. A wire carrying a current of 10.0 A is inserted along the axis of the solenoid. What is the magnitude of the magnetic field at a point \(1.00 \mathrm{~cm}\) from the axis?

Two long, straight wires are parallel to each other. The wires carry currents of different magnitudes. If the amount of current flowing in each wire is doubled, the magnitude of the force between the wires will be a) twice the magnitude of the original force. b) four times the magnitude of the original force. c) the same as the magnitude of the original force. d) half of the magnitude of the original force.

A long, straight cylindrical tube of inner radius \(a\) and outer radius \(b\), carries a total current \(i\) uniformly across its cross section. Determine the magnitude of the magnetic field from the tube at the midpoint between the inner and outer radii.
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