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Chapter 24: Problem 16

A researcher would like to perform an experiment in zero magnetic field, which means that the field of the earth must be canceled. Suppose the experiment is done inside a solenoid of diameter \(1.0 \mathrm{m},\) length \(4.0 \mathrm{m},\) with a total of 5000 turns of wire. The solenoid is oriented to produce a field that opposes and exactly cancels the \(52 \mu\) T local value of the earth's field. What current is needed in the solenoid's wire?

Short Answer

Expert verified
The current needed in the solenoid's wire to cancel the earth's magnetic field is approximately 1.32 Amperes.

Step by step solution

01

Formulate the Given Parameters

The given parameters for this problem include: \nThe magnetic field of the earth, \(B_{earth} = 52\) 渭T = \(52 x 10^{-6}\) T (converted from 渭T to T), \nThe solenoid coil's total number of turns, \(N = 5000\),\nAnd, the length of the solenoid, \(L = 4.0\) m.
02

Apply Ampere's Law

According to Ampere's Law, the magnetic field \(B\) in the solenoid is given by the formula: \[B = \frac{渭_0IN}{L}\]\nwhere \(I\) is the current flowing through the wire, \(N\) is the total number of turns, \(L\) is the length of the solenoid, and \(渭_0\) is the permeability of free space (equal to \(4蟺 x 10^{-7} Tm/A\)). From the problem statement, we know that \(B\) should equal \(B_{earth}\) in order to cancel out the Earth's magnetic field.
03

Solve for the Current

To solve for the current, \(I\), we rearrange the equation from step 2 to get: \n\[I = \frac{B_{earth}L}{渭_0N}\]\nWe then substitute the given values: \n\[I = \frac{(52 x 10^{-6}T)(4.0m)}{(4蟺 x 10^{-7}Tm/A)(5000)}\]
04

Calculate and Express Your Answer

After executing the calculation, we get: \n\[I 鈮� 1.32 A\]\nThe current needed in the solenoid's wire is approximately 1.32 Amperes.

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Key Concepts

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

Magnetic Field
A magnetic field is an invisible force produced by moving electric charges. It surrounds magnetic materials and currents in wires. In the context of the problem, we're looking at the Earth's magnetic field, which is quite weak, measuring about 52 渭T (microteslas) locally.
This magnetic field is what we need to counteract using a solenoid. Understanding magnetic fields is key to many applications, including compasses and motors. When dealing with fields, we measure the strength in teslas (T). For context, 1 tesla is a very strong magnetic field.
  • Definition: A region around a magnetic material or a moving electric charge within which the force of magnetism acts.
  • Magnetic Field of the Earth: Generally ranges from 25 to 65 渭T depending on where you are on Earth.
Recognizing how magnetic fields interact helps us in designing devices that rely on magnetism.
Solenoid
A solenoid is a coil of wire that creates a magnetic field when an electric current passes through it. The solenoid in this problem is quite large, with 5000 turns of wire, creating a strong magnetic field that can exactly cancel the Earth's field.
The design of the solenoid is crucial. It鈥檚 cylindrical, allowing the field lines to focus inside the coil. This shape and the number of turns determine the strength and uniformity of the magnetic field produced.
  • Function: Converts electrical energy into magnetic energy when current flows.
  • Applications: Used in electromagnets, inductors, antennas, and sensors.
The length and number of turns of the solenoid are essential in optimizing the field it produces.
Permeability of Free Space
The permeability of free space, denoted as \(渭_0\), is a constant that expresses how much resistance is encountered when forming a magnetic field in a vacuum. Its value is approximately \(4蟺 \times 10^{-7} \ Tm/A\).
This constant plays a significant role in Ampere's Law, which helps us determine the strength of the magnetic field within a solenoid. In our problem, to cancel out the Earth's magnetic field, the permeability is crucial for calculating the required current in the solenoid.
  • Definition: The measure of the ability of a material, in this case, free space, to support the formation of a magnetic field.
  • Significance: Provides the relationship between magnetic field strength and current in ampere's circuits.
Understanding permeability is vital for designing electromagnets and in various physics applications.

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Most popular questions from this chapter

An electromagnetic flow meter applies a magnetic field of \(0.20 \mathrm{T}\) to blood flowing through a coronary artery at a speed of \(15 \mathrm{cm} / \mathrm{s} .\) What force is felt by a chlorine ion with a single negative charge?

Two concentric current loops lie in the same plane. The smaller loop has a radius of \(3.0 \mathrm{cm}\) and a current of \(12 \mathrm{A}\). The bigger loop has a current of 20 A. The magnetic field at the center of the loops is found to be zero. What is the radius of the bigger loop?

The moving ions can be thought of as a current loop, and it will produce its own magnetic field. The direction of this field at the center of the particles' circular orbit is A. In the same direction as the spectrometer's magnetic field. B. Opposite the direction of the spectrometer's magnetic field.

A 1.0 -m-long, 1.0 -mm-diameter copper wire carries a current of \(50.0 \mathrm{A}\) to the east. Suppose we create a magnetic field that produces an upward force on the wire exactly equal in magnitude to the wire's weight, causing the wire to "levitate." What are the field's direction and magnitude?

II A square current loop \(5.0 \mathrm{cm}\) on each side carries a \(500 \mathrm{mA}\) current. The loop is in a \(1.2 \mathrm{T}\) uniform magnetic field. The axis of the loop, perpendicular to the plane of the loop, is \(30^{\circ}\) away from the field direction. What is the magnitude of the torque on the current loop?
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