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Chapter 20: Problem 5

An ideal solenoid with a current of 1.5 A has a radius of \(3.0 \mathrm{~cm}\) and a turn density of 250 turns \(/ \mathrm{m}\). (a) What is the magnetic flux (due to its own field) through only one of its loops at its center? (b) What current would be required to double the flux value in part (a)?

Short Answer

Expert verified
(a) \( \Phi = 1.06 \times 10^{-5} \; \mathrm{Wb} \). (b) \( I' = 3.0 \; \mathrm{A} \).

Step by step solution

01

Understand the Formula for Magnetic Field Inside a Solenoid

The magnetic field inside an ideal solenoid is given by the equation \( B = \mu_0 n I \), where \( \mu_0 \) is the permeability of free space (\(4\pi \times 10^{-7} \; \mathrm{T \cdot m/A}\)), \( n \) is the turn density (turns per meter), and \( I \) is the current.
02

Calculate the Magnetic Field

Substitute the given values: \( n = 250 \; \mathrm{turns/m} \) and \( I = 1.5 \; \mathrm{A} \) into the formula to find \( B \). \[ B = \mu_0 n I = (4\pi \times 10^{-7}) \times 250 \times 1.5 \]
03

Calculate the Area of One Loop

The area \( A \) of one loop with radius \( r = 3.0 \; \mathrm{cm} = 0.03 \; \mathrm{m} \) is found using \( A = \pi r^2 \). \[ A = \pi \times (0.03)^2 \]
04

Calculate the Magnetic Flux Through One Loop

The magnetic flux \( \Phi \) through one loop is given by \( \Phi = B \cdot A \). Use the values of \( B \) from Step 2 and \( A \) from Step 3.\[ \Phi = B \times A \]
05

Determine the Current Required to Double the Flux

The flux needs to be doubled, so set up the equation \( 2\Phi = B'\cdot A \) where \( B' = \mu_0 n I' \) for the new current \( I' \). Solve for \( I' \) as follows:\[ 2(B \cdot A) = \mu_0 n I' \cdot A \]\[ 2B = \mu_0 n I' \]Solve for \( I' \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solenoid
A solenoid is a coil of wire that is cylindrical in shape, and it is often used to create a uniform magnetic field within its interior. It consists of multiple loops, or turns, often closely wound together. Solenoids are very useful in physics and engineering applications owing to their ability to efficiently create magnetic fields. When an electric current passes through the turns of the solenoid, it generates a magnetic field that is concentrated within the coil. Because of this concentrated direction, solenoids can be used in a variety of electromagnetic devices such as inductors, transformers, or electromagnets.
Magnetic Field
The magnetic field inside a solenoid is a crucial concept because the solenoid's main purpose is to generate a strong, easily directed magnetic field. The strength of this magnetic field is determined by several factors, including the current running through the solenoid, the number of turns per unit length (turn density), and the intrinsic properties of the coil material and space around it. Mathematically, the magnetic field inside an ideal solenoid can be described by the equation \( B = \mu_0 n I \), where \( B \) is the magnetic field, \( \mu_0 \) is the permeability of free space, \( n \) is the turn density, and \( I \) is the current. The field is generally uniform inside the solenoid but weakens rapidly outside of it.
Turn Density
Turn density, also known as the number of windings per unit length, is denoted by \( n \) and is expressed in turns per meter. It is a vital factor affecting the strength of the magnetic field inside a solenoid. A higher turn density means more loops are packed in a given length of the solenoid, amplifying the magnetic field generated by each increment of current. This characteristic of being directly proportional ensures that for a constant current and material, increasing the turn density will correspondingly boost the magnetic field inside the solenoid. Henceforth, engineers and physicists optimize the turn density to tweak the magnetic field strength required for specific applications.
Permeability of Free Space
The permeability of free space, represented by \( \mu_0 \), is a constant that quantifies the ability of a vacuum to support the formation of a magnetic field. It is an intrinsic property of the vacuum affecting how the magnetic field propagates through space. Numerically, it is known to be \( 4\pi \times 10^{-7} \, \mathrm{T \cdot m/A} \). In the equation \( B = \mu_0 n I \), \( \mu_0 \) is a critical factor influencing the strength of the magnetic field within a solenoid. Understanding permeability is fundamental for predicting how easily a material, or even space itself, can be magnetized when exposed to a magnetic field, which in turn aids in designing and implementing electromagnetic systems efficiently.

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Most popular questions from this chapter

A solenoid of length \(40.0 \mathrm{~cm}\) is made of 10000 circular coils. It carries a steady current of 12.0 A. Near its center is placed a small, flat, circular metallic coil of 200 circular loops, each with a radius of \(2.00 \mathrm{~mm}\). This small coil is oriented so that it receives half of the maximum magnetic flux. A switch is opened in the solenoid circuit and its current drops to zero in \(25.0 \mathrm{~ms}\). (a) What was the initial flux through the small coil? (b) Determine the average induced emf in the small coil during the \(25.0 \mathrm{~ms}\). (c) If you look along the long axis of the solenoid so that the initial 12.0 A current is clockwise, determine the direction of the induced current in the small inner coil during the time the current drops to zero. (d) During the \(25.0 \mathrm{~ms}\), what was the average current in the small coil, assuming it has a resistance of \(0.15 \Omega ?\)

A circular loop (radius of \(20 \mathrm{~cm}\) ) is in a uniform magnetic field of \(0.15 \mathrm{~T}\). What angle(s) between the normal to the plane of the loop and the field would result in a flux with a magnitude of \(1.4 \times 10^{-2} \mathrm{~T} \cdot \mathrm{m}^{2} ?\)

(a) In May 2008 , the United States successfully landed a spacecraft named Phoenix near the northern polar regions of Mars. Immediately upon landing, the craft sent a message indicating that all had gone well. Using the astronomical data in the appendix of this book, determine the shortest amount of time it took this signal to reach the Earth. (b) If the Phoenix transmitter sent out spherical electromagnetic waves with a power of \(100 \mathrm{~W}\), how many watts per square meter would arrive at the Earth, assuming that Mars was in its closest location to Earth? (c) A radio signal was sent to a deep space probe traveling in the plane of the solar system. Earth received a response 3.5 days later. Assuming the probe computers took 4.5 hours to process the signal instructions and to send out the return message, was the probe within the solar system? (Assume a solar system radius of about 40 times the Earth-Sun distance.)

When the magnetic flux through a single loop of wire increases by \(30 \mathrm{~T} \cdot \mathrm{m}^{2},\) an average current of \(40 \mathrm{~A}\) is induced in the wire. Assuming that the wire has a resistance of \(2.5 \Omega,\) (a) over what period of time did the flux increase? (b) If the current had been only \(20 \mathrm{~A},\) how long would the flux increase have taken?

A flat coil of copper wire consists of 100 loops and has a total resistance of \(0.500 \Omega\). The coil diameter is \(4.00 \mathrm{~cm}\) and it is in a uniform magnetic field pointing toward you (out the page). The coil orientation is in the plane of the page. It is then pulled to the right (without rotating) until it is completely out of the field. (a) What is the direction of the induced current in the coil: (1) clockwise, (2) counterclockwise, or (3) there is no induced current? (b) During the time the coil leaves the field, an average induced current of \(20.0 \mathrm{~mA}\) is measured. What is the average induced emf in the coil? (c) If the field strength is \(5.50 \mathrm{mT}\), how much time did it take to pull the coil out?
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