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Chapter 13: Problem 1924

In a mass spectrometer used for measuring the masses of ions, the ions are initially accelerated by an ele. potential \(\mathrm{V}\) and then made to describe semicircular paths of radius \(\mathrm{r}\) using a magnetic field \(\mathrm{B}\). If \(\mathrm{V}\) and \(\mathrm{B}\) are kept constant, the ratio [(Charge on the ion) / (mass of the ion)] will be proportional to. (a) \(\left(1 / r^{2}\right)\) (b) \(r^{2}\) (c) \(\mathrm{r}\) (d) \((1 / \mathrm{r})\)

Short Answer

Expert verified
The ratio (charge on the ion) / (mass of the ion) is proportional to \((1/r)\).

Step by step solution

01

Understand the working principle of mass spectrometer

: In a mass spectrometer, ions are initially accelerated by an electric potential V, and then they move in a magnetic field B along semicircular paths of radius r. The movement of the ions is governed by the Lorentz force formula, which can be applied to find the relationship between charge, mass, and radius.
02

Apply Lorentz force formula

: The Lorentz force acting on the ion can be given as: \(F_{L} = q(vB)\), where \(F_L\) is the Lorentz force, \(q\) is the charge of the ion, \(v\) is the velocity of the ion, and \(B\) is the magnetic field. Also, the centripetal force acting on the ion can be given as: \(F_{c} = \frac{m v^{2}}{r}\), where \(F_c\) is the centripetal force, \(m\) is the mass of the ion, and \(r\) is the radius of the semicircular path. Since the ion is moving in a semicircular path, the centripetal force and Lorentz force acting on the ion must be equal. Hence, we have: \(q(vB) = \frac{m v^{2}}{r}\)
03

Express velocity in terms of electric potential

: The initial acceleration of the ion under the influence of electric potential can be given as: \(qV = \frac{1}{2}m v^{2}\), where \(V\) is the electric potential. From this equation, the velocity of the ion can be expressed in terms of electric potential, mass, and charge: \(v = \sqrt{2qV/m}\)
04

Substitute the value of velocity in Lorentz force equation

: Now, we will substitute the velocity from the electric potential equation into the Lorentz force and centripetal force equation: \(qB\sqrt{2qV/m} = \frac{m (2qV/m)}{r}\)
05

Simplify the equation and find the ratio

: By simplifying the above equation, we can find the ratio (charge on the ion)/(mass of the ion) is proportional to: \(\frac{q}{m} = \frac{1}{rB}\) Thus, the ratio (charge on the ion)/(mass of the ion) is inversely proportional to the radius. Therefore, the correct answer is: (d) \((1/r)\)

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Most popular questions from this chapter

An electron having mass \(9 \times 10^{-31} \mathrm{~kg}\), charge $1.6 \times 10^{-19} \mathrm{C}\( and moving with a velocity of \)10^{6} \mathrm{~m} / \mathrm{s}$ enters a region where mag. field exists. If it describes a circle of radius \(0.10 \mathrm{~m}\), the intensity of magnetic field must be Tesla (a) \(1.8 \times 10^{-4}\) (b) \(5.6 \times \overline{10^{-5}}\) (c) \(14.4 \times 10^{-5}\) (d) \(1.3 \times 10^{-6}\)

Graph of force per unit length between two long parallel current carrying conductors and the distance between them (a) Straight line (b) Parabola (c) Ellipse (d) Rectangular hyperbola

A proton and an particle are projected with the same kinetic energy at right angles to the uniform mag. field. Which one of the following statements will be true. (a) The \(\alpha\) - particle will be bent in a circular path with a small radius that for the proton. (b) The radius of the path of the \(\alpha\) - particle will be greater than that of the proton. (c) The \(\alpha\) - particle and the proton will be bent in a circular path with the same radius. (d) The \(\alpha\) - particle and the proton will go through the field in a straight line.

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If a magnet of pole strength \(\mathrm{m}\) is divided into four parts such that the length and width of each part is half that of initial one, then the pole strength of each part will be (a) \((\mathrm{m} / 4)\) (b) \((\mathrm{m} / 2)\) (c) \((\mathrm{m} / 8)\) (d) \(4 \mathrm{~m}\)
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