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Q. 68

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Physics for Scientists and Engineers: A Strategic Approach with Modern Physics
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Short Answer

A Hall-effect probe to measure magnetic field strengths needs to be calibrated in a known magnetic field. Although it is not easy to do, magnetic fields can be precisely measured by measuring the cyclotron frequency of protons. A testing laboratory adjusts a magnetic field until the proton's cyclotron frequency is . At this field strength, the Hall voltage on the probe is when the current through the probe is . Later, when an unknown magnetic field is measured, the Hall voltage at the same current is . What is the strength of this magnetic field?

The strength of the magnetic field is determined as

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Step by Step Solution

 Step :1 Introduction

The area of space near a magnetic body or a current-carrying body where magnetic forces caused by the body or current can be detected. The frequency of a charged particle traveling perpendicular to the direction of a uniform magnetic field B is known as the cyclotron frequency or gyrofrequency (constant magnitude and direction). The given is the adjustment of a magnetic field until the proton's cyclotron frequency is

The objective is to find the strength of the magnetic field

Step :2 Cyclotron motion 

The cyclotron motion is defined as a particle traveling in a uniform circular motion perpendicular to the magnetic field at a constant speed. The magnetic field causes the circular motion to produce a circular frequency, which is given by equation in the form

Where is the particle's charge, is the magnetic field, and m is the particle's mass. The frequency is determined by the magnetic field rather than the particle's velocity, as indicated by equation . We rewrite equation so that has the form

At a frequency of , the Hall voltage . As a result, we may use this frequency to determine the initial maegtic field B1. Fill in the values for from the equation.

Step :3 Potential difference 

The steady-state potential difference between the two surfaces of a conductor is known as the Hall voltage. Equation in the form relates the Hall voltage to the current inside the conductor and the magnetic field.

Where n is the charge density, is the conductor thickness, is the electron charge, and is the applied magnetic field. The Hall voltage is exactly proportional to the magnetic field, as stated by equation .

We can get an equation for two instants between the Hall voltage and the magnetic field in the form when the current for both probes is the same.

We can calculate at by multiplying 1H=0.543 mV by .

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