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Chapter 19: Problem 85

The intrinsic magnetic dipole moment of the electron has magnitude $9.3 \times 10^{-24} \mathrm{A} \cdot \mathrm{m}^{2} .$ In other words, the electron acts as though it were a tiny current loop with $N I A=9.3 \times 10^{-24} \mathrm{A} \cdot \mathrm{m}^{2} .$ What is the maximum torque on an electron due to its intrinsic dipole moment in a \(1.0-T\) magnetic field?

Short Answer

Expert verified
Answer: The maximum torque on an electron due to its intrinsic dipole moment in a 1.0-T magnetic field is \(9.3 \times 10^{-24} N\cdot m\).

Step by step solution

01

Write the torque formula related to magnetic dipole moment and magnetic field

We know that the torque 蟿 acting on a magnetic dipole of magnetic moment 渭 in a magnetic field B is given by 蟿 = 渭 * B * sin 胃, where 胃 is the angle between the magnetic moment and the magnetic field.
02

Find the maximum torque

To find the maximum torque, we need the maximum value of sin 胃, which is 1 when 胃 = 90 degrees. Therefore, the maximum torque is given by 蟿_max = 渭 * B.
03

Substitute the given values and calculate the maximum torque

Given, intrinsic magnetic dipole moment of the electron (渭) = \(9.3 \times 10^{-24} A\cdot m^2\) and magnetic field (B) = 1.0 T. Substitute these values into the formula: 蟿_max = 渭 * B = \((9.3 \times 10^{-24} A\cdot m^2) * (1.0 T)\). Hence, 蟿_max = \(9.3 \times 10^{-24} N\cdot m\). Thus, the maximum torque on an electron due to its intrinsic dipole moment in a 1.0-T magnetic field is \(9.3 \times 10^{-24} N\cdot m\).

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Most popular questions from this chapter

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?
In a certain region of space, there is a uniform electric field \(\overrightarrow{\mathbf{E}}=2.0 \times 10^{4} \mathrm{V} / \mathrm{m}\) to the east and a uniform magnetic field \(\overline{\mathbf{B}}=0.0050 \mathrm{T}\) to the west. (a) What is the electromagnetic force on an electron moving north at \(1.0 \times 10^{7} \mathrm{m} / \mathrm{s} ?(\mathrm{b})\) With the electric and magnetic fields as specified, is there some velocity such that the net electromagnetic force on the electron would be zero? If so, give the magnitude and direction of that velocity. If not, explain briefly why not.
Find the magnetic force exerted on a proton moving east at a speed of $6.0 \times 10^{6} \mathrm{m} / \mathrm{s}\( by a horizontal magnetic ficld of \)2.50 \mathrm{T}$ directed north.
A straight wire segment of length \(25 \mathrm{cm}\) carries a current of $33.0 \mathrm{A}$ and is immersed in a uniform external magnetic field. The magnetic force on the wire segment has magnitude \(4.12 \mathrm{N} .\) (a) What is the minimum possible magnitude of the magnetic field? (b) Explain why the given information enables you to calculate only the minimum possible field strength.
A uniform magnetic ficld points north; its magnitude is 1.5 T. A proton with kinetic energy \(8.0 \times 10^{-13} \mathrm{J}\) is moving vertically downward in this field. What is the magnetic force acting on it?
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