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Chapter 19: Problem 70

A solenoid of length \(0.256 \mathrm{m}\) and radius \(2.0 \mathrm{cm}\) has 244 turns of wire. What is the magnitude of the magnetic field well inside the solenoid when there is a current of \(4.5 \mathrm{A}\) in the wire?

Short Answer

Expert verified
Answer: The magnitude of the magnetic field inside the solenoid is approximately 5.39 x 10^{-3} T.

Step by step solution

01

Identify the given values

We are given the following values: - Length of the solenoid \((L) = 0.256 \,\text{m}\) - Radius of the solenoid \((R) = 2.0 \,\text{cm} = 0.020 \,\text{m}\) - Number of turns of wire \((n) = 244\) - Current in the wire \((I) = 4.5 \,\text{A}\)
02

Calculate the number of turns per unit length

To find the magnetic field inside the solenoid, we first need to find the number of turns per unit length \((n')\). This value is found by dividing the number of turns by the length of the solenoid: \(n' = \frac{n}{L}\) Plugging in the given values, we find: \(n' = \frac{244}{0.256 \,\text{m}} \approx 953.13 \, \text{turns/m}\)
03

Apply the formula for the magnetic field inside a solenoid

The formula for the magnetic field inside a solenoid is given by: \(B = \mu_0 n' I\) Where \(B\) is the magnetic field, \(\mu_0\) is the magnetic permeability of free space, and \(I\) is the current in the wire. The magnetic permeability of free space is a constant equal to \(\mu_0 = 4\pi \times 10^{-7} \, \text{Tm/A}\). Now, we plug in the values: \(B = (4\pi \times 10^{-7} \, \text{Tm/A})(953.13 \, \text{turns/m})(4.5 \,\text{A})\)
04

Calculate the magnetic field

Performing the multiplication, we obtain the magnitude of the magnetic field inside the solenoid: \(B \approx (4\pi \times 10^{-7} \, \text{Tm/A})(953.13 \, \text{turns/m})(4.5 \,\text{A})\) \(B \approx 5.39 \times 10^{-3} \, \text{T}\) Thus, the magnitude of the magnetic field well inside the solenoid when there is a current of \(4.5 \,\text{A}\) in the wire is approximately \(5.39 \times 10^{-3} \, \text{T}\).

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Most popular questions from this chapter

Point \(P\) is midway between two long, straight, parallel wires that run north- south in a horizontal plane. The distance between the wires is $1.0 \mathrm{cm} .\( Each wire carrics a current of \)1.0 \mathrm{A}$ toward the north. Find the magnitude and direction of the magnetic field at point \(P\)
An early cyclotron at Cornell University was used from the 1930 s to the 1950 s to accelerate protons, which would then bombard various nuclei. The cyclotron used a large electromagnet with an iron yoke to produce a uniform magnetic field of \(1.3 \mathrm{T}\) over a region in the shape of a flat cylinder. Two hollow copper dees of inside radius \(16 \mathrm{cm}\) were located in a vacuum chamber in this region. (a) What is the frequency of oscillation necessary for the alternating voltage difference between the dees? (b) What is the kinetic energy of a proton by the time it reaches the outside of the dees? (c) What would be the equivalent voltage necessary to accelerate protons to this energy from rest in one step (say between parallel plates)? (d) If the potential difference between the dees has a magnitude of $10.0 \mathrm{kV}$ each time the protons cross the gap, what is the minimum number of revolutions each proton has to make in the cyclotron?
An electromagnetic flowmeter is used to measure blood flow rates during surgery. Blood containing ions (primarily Na") flows through an artery with a diameter of \(0.50 \mathrm{cm} .\) The artery is in a magnetic field of $0.35 \mathrm{T}\( and develops a Hall voltage of \)0.60 \mathrm{mV}$ across its diameter. (a) What is the blood speed (in \(\mathrm{m} / \mathrm{s}\) )? (b) What is the flow rate (in \(\mathrm{m}^{3} / \mathrm{s}\) ) \(?\) (c) If the magnetic field points west and the blood flow is north, is the top or bottom of the artery at the higher potential?
A straight wire segment of length \(0.60 \mathrm{m}\) carries a current of $18.0 \mathrm{A}$ and is immersed in a uniform external magnetic ficld of magnitude \(0.20 \mathrm{T}\). (a) What is the magnitude of the maximum possible magnetic force on the wire segment? (b) Explain why the given information enables you to calculate only the maximum possible force.
Find the magnetic force exerted on an electron moving vertically upward at a speed of \(2.0 \times 10^{7} \mathrm{m} / \mathrm{s}\) by a horizontal magnetic field of \(0.50 \mathrm{T}\) directed north.
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