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Chapter 40: Problem 74

The precession frequency of the protons in a laboratory NMR spectrometer is \(15.35850 \mathrm{MHz}\). The magnetic moment of the proton is \(1.410608 \cdot 10^{-26} \mathrm{~J} / \mathrm{T}\), while its spin angular momentum is \(0.5272863 \cdot 10^{-34} \mathrm{~J}\) s. Calculate the magnitude of the magnetic field in which the protons are immersed.

Short Answer

Expert verified
Question: Calculate the magnitude of the magnetic field in which the protons are immersed, given the precession frequency of 15.35850 MHz, the magnetic moment of the proton as 1.410608 x 10^(-26) J/T, and the spin angular momentum of the proton as 0.5272863 x 10^(-34) J*s. Answer: The magnitude of the magnetic field in which the protons are immersed is approximately 0.578 T.

Step by step solution

01

Write down the given values

We are given the following values: - Precession frequency, \(\nu = 15.35850 \mathrm{MHz}\) - Magnetic moment of the proton, \(\mu = 1.410608 \times 10^{-26} \mathrm{J/T}\) - Spin angular momentum of the proton, \(s = 0.5272863 \times 10^{-34} \mathrm{J\cdot s}\)
02

Convert the precession frequency to Hz

We need to convert the precession frequency from MHz to Hz: \(\nu = 15.35850 \mathrm{MHz} = 15.35850 \times 10^6 \mathrm{Hz}\)
03

Rearrange the Larmor equation to solve for the magnetic field

The Larmor equation is given by: \(\nu = \frac{\mu B}{s}\) We need to solve for \(B\), the magnetic field magnitude. Rearranging the Larmor equation, we get: \(B = \frac{\nu \cdot s}{\mu}\)
04

Plug in the given values and calculate the magnetic field magnitude

Now, we can plug in the given values to find the magnetic field magnitude: \(B = \frac{(15.35850 \times 10^6 \mathrm{Hz}) \cdot (0.5272863 \times 10^{-34} \mathrm{J\cdot s})}{1.410608 \times 10^{-26} \mathrm{J/T}}\) Computing the values, we get: \(B \approx 0.578 \mathrm{T}\)
05

State the final answer

The magnitude of the magnetic field in which the protons are immersed is approximately \(0.578 \mathrm{T}\).

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Most popular questions from this chapter

Pu decays with a half-life of 24,100 yr via a 5.25 \(\mathrm{MeV}\) alpha particle. If you have a \(1.00 \mathrm{~kg}\) spherical sample of \({ }^{239} \mathrm{Pu},\) find the initial activity in \(\mathrm{Bq} .\)

Using the table of isotopes in Appendix B, calculate the binding energies of the following nuclei. a) \({ }^{7} \mathrm{Li}\) b) \({ }^{12} \mathrm{C}\) c) \({ }^{56} \mathrm{Fe}\) d) \({ }^{85} \mathrm{Rb}\)

A nuclear reaction of the kind \({ }_{2}^{3} \mathrm{He}+{ }_{6}^{12} \mathrm{C} \rightarrow \mathrm{X}+\alpha\) is called a pick-up nuclear reaction. a) Why is it called a pick-up reaction, that is, what is picked up, what picked it up, and where did it come from? b) What is the resulting nucleus X? c) What is the \(\mathrm{Q}\) -value of this reaction? d) Is this reaction endothermic or exothermic?

Determine the decay constant of radium- 226 , which has a half-life of \(1600 \mathrm{yr}\).

You have developed a grand unified theory which predicts the following things about the decay of the proton: (1) protons never get any older, in the sense that their probability of decay per unit time never changes, and (2) half the protons in any given collection of protons will have decayed in \(1.80 \cdot 10^{29}\) yr. You are given experimental facilities to test your theory: A tank containing \(1.00 \cdot 10^{4}\) tons of water and sensors to record proton decays. You will be allowed access to this facility for two years. How many proton decays will occur in this period if your theory is correct?
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