Video: Relating the Energy of an Accelerated Charged Particle to Its Motion through a Uniform Electric Field

An electron is accelerated in a uniform electric field having a strength of 2.00 × 10⁶ V/m. What energy in KeV is given to the electron if it is accelerated through 0.400 m? Over what distance would it have to be accelerated to increase its energy by 50.0 GeV?

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Video Transcript

An electron is accelerated in a uniform electric field having a strength of 2.00 times 10 to the sixth volts per meter. What energy in kiloelectron volts is given to the electron if it is accelerated through 0.400 meters? Over what distance would it have to be accelerated to increase its energy by 50.0 gigaelectron volts?

In this exercise, we have a uniform electric field we can draw like this. And we’ll label this field capital 𝐸 and we’re given the strength of it in units of volts every meter. That is, this number tells us just how many volts of potential difference there are for every meter of the field.

We’re told further that an electron, a charged particle, is accelerated in this electric field. We’re told the electron has moved through the field at distance of 0.400 meters. And in the process of that movement, it’s been given energy. This gain in energy takes place because the electron is being accelerated from right to left towards a positive direction.

To figure out the energy in kiloelectron volts given to this electron, we can use the fact number one that it’s an electron and number two that our field is expressed in units of volts per meter. As we do, let’s recall what an electron volt is after all.

An electron volt is an amount of energy that’s gained by the charge of a single electron that’s moved across a potential difference of one volt. Keeping this in mind, if we go back to our diagram of an electron moving in a uniform electric field, we can say that as this electron moves through the field, the potential difference 𝑉 that it experiences is equal to the electric field strength multiplied by the distance 𝑑 that the electron moves.

To see that this is plausible, let’s check the units. The units on our electric field are volts per meter and the units on our distance are meters. So if we multiply those two values together, we’ll get something with units of volts, potential difference.

This is useful to us because we just read that an electron volt is an amount of energy gained by the charge of a single electron such as we have here when it’s moved across a potential difference of one volt or that to say the number of volts we get when we multiply electric field by distance will be equal to the number of electron volts of energy. That’s helpful to us because we want to give our answer on energy in units of kiloelectron volts.

So let’s go ahead and calculate potential difference by multiplying the electric field strength by the distance our electron has moved through that field. We call that potential difference 𝑉 and it’s equal to 2.00 times 10 to the sixth volts per meter multiplied by the distance the electron travels. And as we pointed out, the units of meters cancel out and we’re left with units of volts. We find a result of 800000 volts or 800 kilovolts.

As we said, this number of volts that our electron moves through is equal to the number of electron volts of energy that it gains through this motion. This means that to write our answer in terms of energy, we simply take 800 kilovolts and replace volts with electron volts. And we’re able to do that because of the definition of electron volts that one electron volt corresponds to an electron moving across a potential difference of one volt. There is a one-to-one relationship.

That’s it for our first question. The answer to it is 800 kiloelectron volts or 800 KeV.

Next, we want to solve for a distance that our electron would have to be accelerated through in order to increase its energy by a given amount, 50.0 giga electron volts. In this part, we’re given information the other way around. We have the energy of our electron and we want to find out how far it must have moved in order to get that energy.

Going back to our relationship that potential difference is equal to electric field times distance, we can get setup for using this relationship in our case. Now, on the left-hand side, we have a potential difference. But we’re not given a potential difference, but rather an energy in this expression.

But as we saw before, this amount of energy and the number of our volts in potential difference are the same. The only thing that changes is the units from electron volts in energy to volts in potential difference. That’s a complicated way of saying that we can replace our potential difference 𝑉 with 50.0 giga volts because this is the potential difference that would correspond to our electron gaining 50.0 giga electron volts of energy.

So 50.0 giga volts is equal to the electric field strength of the electron moves in multiplied by the distance through which the charge moves. That’s the distance we want to solve for. So to do it, let’s divide both sides by the electric field. Since we’re given 𝐸 the electric field strength as 2.00 times 10 to the sixth volts per meter, when we plug in this value, notice what happens to the units. The units of volts cancel out and the units of meters move into the numerator. So we would get a distance in units of meters.

As we do that, as one last step before we punch in our numbers, we’ll convert 50.0 giga or giga is a prefix meaning billion to 50.0 times 10 to the ninth. We’re seeing that 50.0 times 10 to the ninth meters divided by 2.00 times 10 to the sixth is equal to the distance the electron must have travelled to gain this much energy, 50.0 giga electron volts.

When we calculate this fraction, we find that to the three significant figures it’s equal to 25.0 kilometers, that is one long particle acceleration chamber. But that’s what it will take with this electric field in order to give an electron that much energy.

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