An electron with energy equal to K1 = 5.55 x 10^2 eV and an electron with energy equal to K2 =...

Question:

An electron with energy equal to K1 = 5.55{eq}\times {/eq}10{eq}^2 {/eq} eV and an electron with energy equal to K2 = 2.03{eq}\times {/eq}10{eq}^2 {/eq} eV are trapped in a uniform magnetic field and move in circular paths in a plane perpendicular to the magnetic field. What is the ratio (r1/r2) of the radii of their orbits?

Circular Motion of Charged Particle:

The magnetic force on a moving charge due to the external magnetic field is perpendicular to the velocity vector of the particle, which therefore causes the circular path of the particle with a suitable radius depending on the charge, mass and velocity of the particle.

Answer and Explanation: 1

Given Data

  • Kinetic Energy of electron-1, {eq}K_1\ = 5.55\times 10^2\ \text{eV} {/eq}
  • Kinetic Energy of electron-2, {eq}K_2\ = 2.03\times 10^2\ \text{eV} {/eq}

Assuming:

  • Uniform magnetic field as B
  • charge of the electron as e, and mass of the electron as m

Finding the ratio of the radii ({eq}\dfrac{r_1}{r_2}{/eq}) of the orbits of two electrons

{eq}\begin{align} \text{The radius of the circular path of electron-1 with speed as } v_1 \text{, is expressed as:}\\[0.3cm] r_1\ &= \dfrac{m\times v_1}{e\times B}\\[0.3cm] r_1\ &= \dfrac{\sqrt{2\times m\times \dfrac{1}{2}\times m\times v_1^2}}{e\times B}\\[0.3cm] r_1\ &= \dfrac{\sqrt{2\times m\times K_1}}{e\times B}\tag{1}\\[0.3cm] \text{Similarly for electron-2 with speed as } v_2 \text{ , the radius is expressed as:}\\[0.3cm] r_2\ &= \dfrac{m\times v_2}{e\times B}\\[0.3cm] r_2\ &= \dfrac{\sqrt{2\times m\times \dfrac{1}{2}\times m\times v_2^2}}{e\times B}\\[0.3cm] r_2\ &= \dfrac{\sqrt{2\times m\times K_2}}{e\times B}\tag{2}\\[0.3cm] \text{From Equations-1 and 2:}\\[0.3cm] \dfrac{r_1}{r_2}\ &= \sqrt{\dfrac{K_1}{K_2}}\\[0.3cm] \dfrac{r_1}{r_2}\ &= \sqrt{\dfrac{5.55\times 10^2\ \text{eV}}{2.03\times 10^2\ \text{eV}}}\\[0.3cm] \dfrac{r_1}{r_2}\ &\approx 1.65 \end{align} {/eq}


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Magnetic Forces & Fields Practice Problems

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Chapter 18 / Lesson 7
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Magnetism and electricity are closely related. When current flows in a wire, a magnetic field is generated. In this lesson, we will explore how magnetic fields can exert forces on parallel current-carrying wires.


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