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An electron is to be accelerated in a uniform electric field having a strength of $2.00 \times 10^6 \textrm{ V/m}$. (a) What energy in keV is given to the electron if it is accelerated through 0.400 m? (b) Over what distance would it have to be accelerated to increase its energy by 50.0 GeV?
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Final Answer
  1. $8.00 \times 10^2 \textrm{ keV}$
  2. $25.0 \textrm{ km}$
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OpenStax College Physics Solution, Chapter 19, Problem 23 (Problems & Exercises) (2:44)

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Video Transcript
This is College Physics Answers with Shaun Dychko. The size of the change in kinetic energy of this electron is equal to the size of the change in its potential energy which will be its charge times V and I am putting these absolute value bars here to say, let’s just ignore negative signs. Now, the potential difference for a uniform electric field is the electric field multiplied by the distance travelled through the field and we substitute that in for V and this is the same formula we have for the potential difference between two large conducting parallel plates and the reason it applies is because the electric field between two large conducting parallel plates is uniform, okay. So… we have a change in energy then is charge times electric field times distance. So, we are dealing with electrons, so the charge is the elementary charge of 1.6 times ten to the minus 19 coulombs times two times ten to the six volts per meter electric field strength times 0.400 meter distance travelled and then we are asked to express our answers in unit of kilo-electron volt. So, this will give us joules and we have this conversion factor to go from joule to electron-volts, one electron-volt for every 1.60 times ten to the minus nineteen joules and then we convert that in to kilo-electron volt by multiplying by one kilo-electron volt for every 1000 electron-volts and this works out to 800 kilo-electron volts and perhaps I should write the answer in scientific notation to indicate that there are two significant figures, … oh sorry three significant figures lets just say, there. Okay. Now, part B asks us over what distance would it have to be accelerated in this field to gain an energy of 50 Giga-electron volt. So, we have q E d equals the change in kinetic energy, that’s what we said up here and we will solve this for d by dividing both sides by q times E. So, d is the change in energy over E times q, and change in energy is 50 Giga-electron volt which we have to convert into joules in order to use in this formula so we multiply by one times ten to the nine electron-volt for every Giga-electron volt and then multiply by 1.60 times ten to the minus 19 joules for every electron-volt and we are left with joules and we divide that by electric field times the only true charge and we end up with 25.0 kilometres, is the distance over which electron would need to be accelerate by this field to get an energy of 50 giga-electron volt.