KAHIBARO
Table of Contents
Charged Particles Gain Energy in an Electric Field
Electric fields are one of the most direct ways to speed up charged particles. If a charged particle is placed in an electric field, the field exerts a force on it. That force can change the particle’s motion and increase its kinetic energy.
For a particle with charge $q$ in an electric field $\vec{E}$, the force is
$$
\vec{F} = q\vec{E}.
$$
A positive charge is pushed in the direction of the electric field. A negative charge is pushed in the opposite direction. Because of this, electric fields can accelerate both positively and negatively charged particles, but in opposite directions.
Important rule:
$$
\vec{F} = q\vec{E}
$$
A charged particle accelerates when an electric field acts on it. The direction of acceleration depends on the sign of the charge.
From Force to Acceleration
Once a force acts on the particle, Newton’s second law connects that force to acceleration:
$$
\vec{F} = m\vec{a}.
$$
Combining this with the electric force gives
$$
m\vec{a} = q\vec{E},
$$
so
$$
\vec{a} = \frac{q}{m}\vec{E}.
$$
This shows an important idea in accelerator physics. The acceleration depends on the charge-to-mass ratio, $q/m$. A particle with larger charge or smaller mass responds more strongly to the same electric field.
Electrons, for example, are much lighter than protons, so they can be accelerated much more easily by the same field.
Key relation:
$$
\vec{a} = \frac{q}{m}\vec{E}
$$
For the same electric field, particles with larger $q/m$ accelerate more strongly.
Electric Potential Difference and Energy Gain
In particle accelerators, the most useful idea is often not just force, but energy gain. When a charged particle moves through a potential difference $\Delta V$, its electric potential energy changes, and this change appears as kinetic energy if losses are negligible.
The work done by the electric field is
$$
W = q\Delta V,
$$
in magnitude when the particle moves through a potential difference $\Delta V$. This work becomes a change in kinetic energy:
$$
\Delta K = q\Delta V.
$$
If a particle starts from rest, then its final kinetic energy is approximately
$$
K = q\Delta V.
$$
This is one of the central formulas of electric-field acceleration.
If an electron moves through a potential difference of $1000 \, \text{V}$, it gains $1000 \, \text{eV}$ of kinetic energy. If a proton moves through the same potential difference, it also gains $1000 \, \text{eV}$. The energy gain depends on charge and voltage, not on mass.
Most important energy formula:
$$
\Delta K = q\Delta V
$$
A particle with charge $q$ gains kinetic energy when it passes through a potential difference $\Delta V$.
The Electron Volt
Because particle energies are often very small in joules, accelerator physics commonly uses the electron volt, abbreviated eV.
One electron volt is the energy gained by a particle with charge equal to the magnitude of the electron charge when it moves through a potential difference of $1 \, \text{V}$.
$$
1 \, \text{eV} = 1.602 \times 10^{-19} \, \text{J}.
$$
Larger units are also common.
| Unit | Meaning | In electron volts |
|---|---|---|
| eV | electron volt | $1$ |
| keV | kilo electron volt | $10^3$ eV |
| MeV | mega electron volt | $10^6$ eV |
| GeV | giga electron volt | $10^9$ eV |
| TeV | tera electron volt | $10^{12}$ eV |
If a proton is accelerated through $2 \, \text{MV}$, its kinetic energy gain is $2 \, \text{MeV}$.
Uniform Electric Field Between Plates
A simple way to create electric-field acceleration is to place two conducting plates at different electric potentials. This produces an electric field between them, approximately uniform if edge effects are small.
If the plate separation is $d$ and the potential difference is $\Delta V$, then the field magnitude is approximately
$$
E = \frac{\Delta V}{d}.
$$
A charged particle entering this region experiences a force and accelerates across the gap.
For a positive particle, the force is in the same direction as $\vec{E}$. For a negative particle, the force is opposite to $\vec{E}$.
Why Accelerators Use Gaps
In practical accelerators, particles are often accelerated when they cross specially designed regions called accelerating gaps. Across a gap, there is a potential difference, and therefore an electric field. As the particle passes through, it gains energy.
This idea is extremely important because a particle can pass through many such accelerating regions one after another, gaining energy each time. The total energy gain is then the sum of the gains from each stage.
If a particle gains energy $q\Delta V$ in one gap, then after passing through $N$ identical gaps the total gain is
$$
\Delta K_{\text{total}} = N q \Delta V.
$$
This is the basic principle behind many linear accelerators.
Repeated acceleration principle:
If a particle crosses several accelerating gaps, the energy gains add:
$$
\Delta K_{\text{total}} = \sum q\Delta V
$$
For identical stages,
$$
\Delta K_{\text{total}} = N q\Delta V
$$
Example of Energy Gain
Suppose a proton, with charge $q = +e$, moves through a potential difference of $5000 \, \text{V}$.
Its kinetic energy gain is
$$
\Delta K = q\Delta V = e(5000 \, \text{V}) = 5000 \, \text{eV} = 5 \, \text{keV}.
$$
If instead an alpha particle with charge $+2e$ moves through the same potential difference, then
$$
\Delta K = 2e(5000 \, \text{V}) = 10000 \, \text{eV} = 10 \, \text{keV}.
$$
So a larger charge means a larger energy gain for the same voltage.
Relation Between Energy and Speed
Electric-field acceleration gives energy directly. The resulting speed depends on the particle’s mass.
For nonrelativistic motion,
$$
K = \frac{1}{2}mv^2.
$$
If a particle starts from rest and gains kinetic energy $q\Delta V$, then
$$
q\Delta V = \frac{1}{2}mv^2.
$$
Solving for speed gives
$$
v = \sqrt{\frac{2q\Delta V}{m}}.
$$
This shows that lighter particles reach higher speeds for the same energy gain. However, at very high speeds, relativistic effects become important, and this simple formula is no longer accurate.
For particles starting from rest, in the nonrelativistic limit:
$$
q\Delta V = \frac{1}{2}mv^2
$$
and therefore
$$
v = \sqrt{\frac{2q\Delta V}{m}}
$$
A Simple Picture of an Accelerating Gap
As the positive particle crosses the gap, the electric field does work on it, and its kinetic energy increases. A negative particle could also be accelerated, but the electrode polarities would need to be arranged appropriately.
Limits of Electric-Field Acceleration
Although electric fields are excellent for giving particles energy, there are practical limits. Very large electric fields can cause electrical breakdown in materials or in residual gas. This means sparks or discharge can occur if the field becomes too strong. Because of this, accelerator design must carefully control voltage, geometry, and vacuum conditions.
Another limitation is that once particles become very fast, especially close to the speed of light, increasing their kinetic energy does not increase their speed very much. Instead, the extra energy mainly increases their momentum. This is why high-energy accelerator physics often focuses on energy rather than speed.
Summary of Core Ideas
Electric-field acceleration is based on the force exerted by an electric field on a charged particle. The force is $q\vec{E}$, and the resulting acceleration is $(q/m)\vec{E}$. More importantly for accelerators, a particle gains kinetic energy when it moves through a potential difference. The energy gain is
$$
\Delta K = q\Delta V.
$$
This makes electric fields a powerful tool for accelerating particles in controlled stages. By arranging repeated accelerating gaps, machines can build up very large particle energies from many smaller voltage steps.