How do you calculate acceleration of a charged particle in an electric field?

You multiply the electric charge by the electric field strength and divide that result by the mass. The formula is a = q * E / m. The product of charge and electric field strength is the force, so F = qE. According to Newton's second law this force can be converted into acceleration. For example, if an individual electron (with a charge of about 1.6 x 10 to the power of -19 C and a mass of about 9.1 x 10 to the power of -31 kg) is placed in an electric field with a strength of 1 N/C, then the acceleration will be about 1.76 x 10 to the power of 11 m/s².

Why is the acceleration of an electron so great?

The acceleration is calculated by dividing the product of charge and field strength by mass. The acceleration is so large, however, because the mass of an electron is extremely small. Even with a relatively low field strength, the acceleration can be many times that of Earth's gravity. In an electric field as weak as 1 N/C, for example, the acceleration of the electron is about twenty billion times that of a freely falling object on Earth.

Does the sign of the charge have an effect on the result?

Yes it does have an effect. Positive charges are accelerated in the direction of the electric field while negative charges like electrons are accelerated in the opposite direction. If the magnitude of charge is equal then the magnitudes of acceleration will be same but the directions will be reversed. When a negative charge is entered, the calculator indicates that the direction is reversed and gives a negative acceleration.

How do you calculate field strength from voltage between two plates?

In a uniform electric field between parallel plates, the voltage can be calculated by dividing the voltage by the distance between the plates: E = V/d. If the voltage between the plates is 200 volts and the distance is 1 cm, then the electric field strength is 20000 N/C. Using the range to calculate the voltage in this tool allows you to calculate the electric field strength and use that result directly for calculations of acceleration.

How much kinetic energy does a particle get?

Kinetic energy is half the product of mass and the square of final velocity. When a charge is accelerated by a voltage V, then the energy gained will also be q x V. This forms the basis for the electronvolt. One electronvolt is the amount of energy that an individual electron gains when it traverses a potential difference of one volt. In this calculator, this energy is shown in electronvolts as the default.

What unit should be used for electric field strength?

The electric field strength should be entered in newtons per coulomb (N/C). This unit is completely equivalent to volts per meter (V/m). Since the two values are numerically equal, 500 N/C would also be 500 V/m. If you only know the voltage and distance between electrodes, use the area of calculating voltage first to find the electric field strength.

Acceleration of a Particle in an Electric Field Calculator

Charge, field and mass

Charge (q)

Electric field (E)

N/C

Mass (m)

Acceleration (a)

Enter a charge, an electric field and a mass, and the calculator finds the acceleration from a = q x E / m. Leave any one of the four boxes blank to solve for it instead. You can also get the field from a voltage, or work out the particle's speed and energy, in the sections below.

Find the field from a voltage

Enter the plate voltage and the gap between the plates and the calculator works out the field for you (E = V / d).

Add speed and energy over time

Enter a starting speed and a time and read the speed, the distance travelled and the kinetic energy the particle picks up.

Please input values into the fields

When charged particles are placed in an electric field they experience a force and accelerate. This calculator allows you to calculate the acceleration from the charge, field strength and mass or vice versa. If any of the four values is missing this unknown can be determined.

You can also calculate the electric field from the voltage between two electrodes and analyze the motion of the particle to determine its velocity, distance traveled, and kinetic energy.

Formula for acceleration of charged particles in an electric field:

When a particle with charge q is placed in an electric field of strength E, it experiences a force due to the electric field. The acceleration of the particle can be calculated from this force using Newton's second law.

a = \dfrac{qE}{m}

In this formula a is the acceleration, q is the charge, E is the field strength and m is the mass. In SI units, the unit of charge is Coulomb, the unit of field strength is Newton per Coulomb and the unit of mass is Kilogram. The unit of calculated acceleration is Meter per second squared.

Derivation of formula.

The force experienced by a charge in an electric field is the electrostatic force and is described by F = qE. This is essentially the same interaction as the Coulomb force between two charges, but here it is expressed using an electric field corresponding to a unit charge.

F = qE

According to Newton's second law of motion, force is equal to mass times acceleration, or F=ma. If you set the two equations for force equal to each other and divide by mass, you get the desired equation.

qE = ma \quad\Longrightarrow\quad a = \dfrac{qE}{m}

So the larger the amount of charge and the stronger the electric field, the greater the acceleration. The heavier a particle is, the smaller its acceleration. The direction of the acceleration is determined by the sign of the charge: positive charges are accelerated in the direction of the electric field, while negative charges like electrons are accelerated in the opposite direction.

Example of calculating acceleration for an electron:

If a single electron is placed in a weak electric field where the force is 1 Newton per coulomb, what will be its acceleration? The charge of an electron is approximately 1.6x10 to the -19 coulombs and the mass is about 9.1x10 to the -31 kilograms.

When you divide the product of charge and electric field strength by mass, you get an acceleration of about 1.76 x 10 to the power of 11 meters per second squared. This value is astonishingly large: The acceleration of an electron is about twenty billion times larger than the gravitational acceleration of a falling object. When a very small amount of charge acts on an extremely small mass, it creates a very large acceleration.

Instructions for using this calculator:

Enter any three of the four input fields (charge, electric field strength, mass, acceleration), and the calculation tool will calculate the value of the missing field. If the field for acceleration is left blank, then the acceleration can be calculated from the other three values. Conversely, you can leave one of the fields (charge, electric field strength, mass) blank and back-calculate the value based on a known acceleration.

For each input field, you can select the units separately. The charge can be given in elementary charges, the mass as rest masses of electrons or protons. Acceleration can be shown in meters per second squared (m/s²) or in g-units. All unit conversions are done automatically by the calculation tool.

Back-calculation of charge, electric field strength and mass:

Since these four quantities are related by the same equation, if three of them are known, then the fourth can be determined. By rearranging the equation a=qE/m, you obtain the following three formulas for back-solving:

q = \dfrac{ma}{E} \qquad E = \dfrac{ma}{q} \qquad m = \dfrac{qE}{a}

The first formula is used to calculate the charge from a measured acceleration. The second formula calculates the electric field strength required to produce this acceleration and the third formula calculates the mass of the particle. If you leave the input box for the desired quantity blank, then the calculator will automatically select the corresponding conversion.

The electric field strength is calculated from the voltage between the electrodes.

The electric field is typically created by two parallel, uniformly charged electrodes with a constant voltage. The electric field between the two electrodes is uniform and its strength is the result of dividing the voltage between the electrodes by the distance between the electrodes.

E = \dfrac{V}{d}

If you expand the voltage calculation area and enter the potential difference and the distance between the electrodes, the tool first calculates the electric field strength and then directly uses this result in the acceleration calculation. The higher the voltage and the smaller the distance between the electrodes, the stronger the electric field is.

For example, the electric field strength across a gap of one centimeter with an applied voltage of 200 volts is about 20,000 newtons per coulomb. This is because 200 divided by 0.01 is 20,000. A charged particle placed in this gap will accelerate according to the formula a=qE/m.

Particle speed, path and energy:

Once the acceleration is determined it can be used to calculate the speed and distance travelled by the particle since it moves with constant acceleration. If the initial velocity is v0, then after a time t the particle will have reached a final velocity of and will have travelled a distance of .

v = v_0 + at \qquad d = v_0 t + \tfrac{1}{2} a t^2

The kinetic energy gained by a particle is half the product of its mass and the square of its velocity. When a charged particle is accelerated through a potential difference, the energy it gains is also equal to the product of its charge and the potential difference. This is the origin of the electronvolt, a unit that represents the amount of energy gained by a single electron when it moves across a potential difference of one volt.

E_k = \tfrac{1}{2} m v^2 = qV

If you extend the calculation area for motion and enter initial velocity and time of movement, the calculator will show final velocity, distance traveled and kinetic energy. In calculations involving particles, kinetic energy is displayed by default in electronvolts.

Units for each physical quantity:

Because the calculator automatically processes units, you can combine different units flexibly. The electric field strength is entered in newtons per coulomb, which however is completely equivalent to volts per meter.

Quantity	Symbol	SI unit
Charge	q	coulomb (C); an electron is 1.6 x 10 to the minus 19 C
Electric field	E	newton per coulomb (N/C) = volt per metre (V/m)
Mass	m	kilogram (kg)
Acceleration	a	metre per second squared (m/s^2)
Voltage	V	volt (V)
Kinetic energy	E_k	joule (J); or electron-volt (eV)

Practical applications:

The technology of controlling the motion of charged particles using an electric field is fundamental to many technologies. Particle accelerators increase the speed of particles by using electrical fields, and early television tubes and oscilloscopes used deflection plates to direct beams of electrons onto a screen.

Inkjet printers give tiny droplets of ink an electric charge and control their direction using an electric field. Mass spectrometers separate ions based on differences in their acceleration in an electric field. Regardless of the application, the basis is always the same simple relationship: a = qE / m.

Common mistakes:

A common mistake is forgetting how small the elementary charge is. Since an electron's charge is about 1.6 x 10 to the negative 19 coulombs, you can convert the unit into "e" and type in the number of elementary charges directly without having to manually type out a long string of zeros.

Also pay attention to the units of electric field strength. Newtons per Coulomb and Volts per meter are completely equivalent. However, if you know voltage and distance, then you first have to divide the voltage by the distance in order to calculate the electric field strength. In this tool you can do that automatically in the Voltage calculation section. Also, for such problems, gravity is usually negligible because the acceleration of charged particles in an electric field is much larger than Earth's gravitational acceleration of 9.8 meters per second squared.

This tool is for learning purposes and solving everyday problems only. For research work, engineering tasks or safety-critical activities you must verify the results against standards and data relevant to your project.

Frequently asked questions

How do you calculate acceleration of a charged particle in an electric field?: You multiply the electric charge by the electric field strength and divide that result by the mass. The formula is a = q * E / m. The product of charge and electric field strength is the force, so F = qE. According to Newton's second law this force can be converted into acceleration. For example, if an individual electron (with a charge of about 1.6 x 10 to the power of -19 C and a mass of about 9.1 x 10 to the power of -31 kg) is placed in an electric field with a strength of 1 N/C, then the acceleration will be about 1.76 x 10 to the power of 11 m/s².
Why is the acceleration of an electron so great?: The acceleration is calculated by dividing the product of charge and field strength by mass. The acceleration is so large, however, because the mass of an electron is extremely small. Even with a relatively low field strength, the acceleration can be many times that of Earth's gravity. In an electric field as weak as 1 N/C, for example, the acceleration of the electron is about twenty billion times that of a freely falling object on Earth.
Does the sign of the charge have an effect on the result?: Yes it does have an effect. Positive charges are accelerated in the direction of the electric field while negative charges like electrons are accelerated in the opposite direction. If the magnitude of charge is equal then the magnitudes of acceleration will be same but the directions will be reversed. When a negative charge is entered, the calculator indicates that the direction is reversed and gives a negative acceleration.
How do you calculate field strength from voltage between two plates?: In a uniform electric field between parallel plates, the voltage can be calculated by dividing the voltage by the distance between the plates: E = V/d. If the voltage between the plates is 200 volts and the distance is 1 cm, then the electric field strength is 20000 N/C. Using the range to calculate the voltage in this tool allows you to calculate the electric field strength and use that result directly for calculations of acceleration.
How much kinetic energy does a particle get?: Kinetic energy is half the product of mass and the square of final velocity. When a charge is accelerated by a voltage V, then the energy gained will also be q x V. This forms the basis for the electronvolt. One electronvolt is the amount of energy that an individual electron gains when it traverses a potential difference of one volt. In this calculator, this energy is shown in electronvolts as the default.
What unit should be used for electric field strength?: The electric field strength should be entered in newtons per coulomb (N/C). This unit is completely equivalent to volts per meter (V/m). Since the two values are numerically equal, 500 N/C would also be 500 V/m. If you only know the voltage and distance between electrodes, use the area of calculating voltage first to find the electric field strength.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Wikipedia: Electric field
Definition of the electric field, the force F = qE on a charge, and the units newton per coulomb and volt per metre.
Wikipedia: Lorentz force
The electric force qE on a charged particle and how it drives the particle's motion.
LibreTexts: Charged Particle in an Electric Field
A particle of charge q and mass m in a field E accelerates at qE/m; the kinematics that follow.