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An electron, that has been accelerated from rest by a potential difference of $250\ \rm V$, enters a region of magnetic field of strength $0.12\ \rm T$. Show that speed of the electron after acceleration is $9.4 \times 10^6\ \rm m s^{–1}.$

For this problem I first tried to find the velocity by rewriting the equation for the force an electron will experience when moving through a magnetic field B. $$F=qvB$$ $$v=\frac{F}{qB}$$ I then attempted to find an equation which I could rewrite to give me $F$ and then substitute it into the equation for velocity. However, I can only find $$E=\frac{F}{q}$$ for an electric field. I know the charge of an electron, but with the information given I cant find the electric field strength.

Any ideas as to which equation I could use?

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2 Answers 2

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You need to observe that magnetic field never does any work on a moving charge so it can't accelerate or decelerate the electron. All the kinetic energy that the electron must have gained must come from the initial accelerating potential. So as the the electron is accelerated through a potential say V we can denote its kinetic energy as $\ \rm eV$. Now also we know that $KE=\frac{1}{2}mv^2$ by equating the two you can obtain the velocity of the accelerated electron.

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  • $\begingroup$ Thanks for the answer. Where do you get $eV$ from? From which expression of kinetic energy is that from? $\endgroup$
    – Pablo
    Dec 2, 2018 at 11:22
  • $\begingroup$ Work done by any electric field on a charge particle is given as (charge)(potential difference between the two points) in this case charge is e and the P.D is V $\endgroup$
    – Aditya Garg
    Dec 2, 2018 at 11:45
  • $\begingroup$ @Pablo eV is the customary unit of energy in particle physics. It's used because it's got such a nice direct relation to the experimental setup: A particle with the charge of 1e (electrons, protons, deuterons, etc.) gains/looses exactly 1eV when it's accelerated by an electric field of 1V. I.e. You want 42eV electrons? Just put 42 volts on your electron gun. You want 100keV X-Rays, apply 100kV to your X-Ray emitter (ok, not precisely because of conversion losses, but you get the idea...). In your question, the potential is 250V, you have electrons, so 250eV. Simple, isn't it? $\endgroup$
    – cmaster - reinstate monica
    Dec 2, 2018 at 12:47
  • $\begingroup$ Magnetic fields certainly do "accelerate or decelerate" charged particles. Perhaps you should rephrase that. $\endgroup$
    – ProfRob
    Dec 2, 2018 at 18:44
  • $\begingroup$ The acceleration I want to talk about here is the change in speed and not velocity ......... Please understand and sorry @Rob Jeffries $\endgroup$
    – Aditya Garg
    Dec 2, 2018 at 19:38
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Since magnetic fields do no work, the electron's kinetic energy is 250 eV, giving the desired speed.

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