Epsilon in Physics Equations (Formula Guide)

In physics, ε (epsilon) does not have one universal meaning. Its definition comes from the equation and topic: it may mean permittivity, engineering strain, or a small dimensionless parameter, among other uses. Before calculating, identify its units, subscript, and assumptions. This guide shows how to choose the right formula, substitute values, and check your result.

As you move into more advanced physics, you will meet the same Greek letter in different subjects. That can make a formula sheet seem clear until you try to use it in a new problem. The key is to treat each epsilon as a symbol whose meaning depends on its equation, not as one fixed quantity.

I have found that a quick check of the topic, units, and subscripts prevents many avoidable errors. The steps below help you identify what epsilon means, use the correct formula, and test whether your answer makes physical sense.

How to identify what epsilon means

Epsilon is a symbol, not a definition. Its meaning comes from the equation, the topic, and any definitions given in your course materials. Start by locating the formula, then check the nearby text, subscripts, and units. These clues help you distinguish permittivity from strain or a small parameter.

In typed text, epsilon may appear as ε or ϵ. These are two forms of the same Greek letter, though a course or author may use one form consistently. A subscript matters: (\varepsilon_0), (\varepsilon_r), and (\varepsilon) can refer to different quantities.

Use this quick process before choosing a value:

• Find the topic: Is the equation about electricity, a material changing shape, or an approximation?
• Check the subscript: The zero in (\varepsilon_0) identifies vacuum permittivity. The (r) in (\varepsilon_r) means relative permittivity.
• Check the units: Units of (\mathrm{F\,m^{-1}}) point to permittivity. Strain and relative permittivity have no units.
• Read the definition: Your textbook or problem may define epsilon in a special way.

If the equation does not define epsilon, use the chapter’s notation list or ask your instructor. Do not assume a familiar-looking symbol means the same thing in every subject.

Epsilon in permittivity formulas

Permittivity describes how a material responds to an electric field. Vacuum permittivity, (\varepsilon_0), is a constant used in electromagnetism, while (\varepsilon) may describe a material. Relative permittivity, (\varepsilon_r), compares a material’s permittivity with that of a vacuum and has no units.

Vacuum permittivity and Coulomb’s law

For two point charges in a vacuum, Coulomb’s law gives the size of the electric force:

[ F=\frac{1}{4\pi\varepsilon_0}\frac{|q_1q_2|}{r^2}. ]

Here, (F) is force in newtons, (q_1) and (q_2) are charges in coulombs, and (r) is the distance between them in metres. The vacuum permittivity is approximately

[ \varepsilon_0 \approx 8.854\times10^{-12}\ \mathrm{F\,m^{-1}}. ]

Use the precision the problem asks for. If it gives a rounded value for (\varepsilon_0), follow that value rather than adding extra digits to your final answer.

Relative and material permittivity

Relative permittivity is defined as

[ \varepsilon_r=\frac{\varepsilon}{\varepsilon_0}. ]

Because it is a ratio of two quantities with the same units, (\varepsilon_r) has no units. Rearranging gives the permittivity of a material:

[ \varepsilon=\varepsilon_r\varepsilon_0. ]

The units of (\varepsilon) are (\mathrm{F\,m^{-1}}). For a linear, isotropic material in a capacitor with the same geometry, capacitance relates to its vacuum value as

[ C=\varepsilon_r C_0, ]

where (C_0) is the capacitance with vacuum between the plates. This simple relation depends on the stated material and geometry assumptions; do not apply it automatically to every capacitor.

A common mistake is to put (\varepsilon_r) into a formula that requires (\varepsilon_0). The result then fails a units check: one is dimensionless, while the other has units of (\mathrm{F\,m^{-1}}).

Epsilon as engineering strain

Engineering strain measures how much an object’s length changes compared with its original length. It is written as (\varepsilon=\Delta L/L_0), where (L_0) is the original length and (\Delta L) is the change. Because both lengths use the same units, strain has no units.

The formula is

[ \varepsilon=\frac{\Delta L}{L_0}, \qquad \Delta L=L-L_0, ]

where (L) is the final length. For example, if a wire starts at (2.00\ \mathrm{m}) and stretches by (0.004\ \mathrm{m}), then

[ \varepsilon=\frac{0.004}{2.00}=0.002. ]

This strain is dimensionless. It may also be reported as a percentage: (0.002\times100=0.2\%). Keep the decimal form or the percentage form clear when using it in another equation.

The sign can carry meaning. If the object lengthens, (\Delta L) and strain are positive under this convention. If it shortens, they are negative. Check the problem’s convention, especially when working with stress and strain graphs.

Epsilon as a small parameter

In an approximation, epsilon may stand for a small, dimensionless quantity. The notation (|\varepsilon|\ll1) means that its magnitude is much less than one for the problem at hand. It does not set one universal cutoff for what counts as “small.”

For example, a formula may use the first terms of an expansion and leave out terms involving (\varepsilon^2) or higher powers. Whether those omitted terms are safe to ignore depends on the size of epsilon and the accuracy needed.

If (\varepsilon=0.01), then (\varepsilon^2=0.0001), which is smaller than (\varepsilon). But that comparison alone does not prove the approximation is accurate enough. Consider the coefficients in the full expression and the allowed error. In a problem or textbook, look for a stated approximation, such as “to first order,” and follow its limits.

Worked examples: choosing the right epsilon

These examples show how context changes the calculation. Each one uses a different meaning of epsilon, so the symbol alone is not enough to select a formula.

A force between charges

For two charges separated in a vacuum, use (\varepsilon_0) in Coulomb’s law. If the problem instead describes a material between the charges, check whether it gives that material’s permittivity and whether the equation is meant for a uniform medium.

Do not swap in (\varepsilon_r) alone. If you need the material permittivity, first calculate (\varepsilon=\varepsilon_r\varepsilon_0), then use the formula that matches the situation.

A capacitor with a dielectric

Suppose a problem gives a capacitor’s vacuum capacitance (C_0) and the relative permittivity of a material between its plates. Under the linear, isotropic material assumption, use (C=\varepsilon_r C_0). Since (\varepsilon_r) has no units, the result has the same units as (C_0), typically farads.

This is a useful example of a units check: multiplying capacitance by a dimensionless number leaves its units unchanged. If your calculation produces units of length or force, revisit the equation or substitution.

A stretched sample

A lab sample has an original length of (50\ \mathrm{mm}) and a final length of (50.5\ \mathrm{mm}). Its change in length is (0.5\ \mathrm{mm}), so

[ \varepsilon=\frac{0.5\ \mathrm{mm}}{50\ \mathrm{mm}}=0.01. ]

The millimetres cancel. The result is a strain of (0.01), or (1\%). This cancellation confirms that strain is dimensionless.

How to check an epsilon calculation

A reliable calculation has a clear definition, suitable units, and assumptions that match the problem. After identifying epsilon, write down what each symbol means before inserting numbers. This small step makes it easier to catch a wrong subscript or a value taken from the wrong topic.

Use this sequence:

1. Write the formula in symbols. Do not begin by entering numbers into a calculator.
2. Define each epsilon. Note whether it is (\varepsilon_0), (\varepsilon_r), strain, or a small parameter.
3. Record units and assumptions. For example, Coulomb’s law shown above is for a vacuum.
4. Convert units. Use consistent units, such as metres and coulombs in the SI form of Coulomb’s law.
5. Substitute values and calculate. Keep enough digits during the calculation, then round the final result as required.
6. Check the dimensions and result. Ask whether the units and the direction or size of the result make sense.

A limiting-case check can help too. In (C=\varepsilon_r C_0), setting (\varepsilon_r=1) gives (C=C_0), as expected for the comparison with vacuum. In the strain formula, no length change gives (\varepsilon=0). If a calculation fails a simple check like these, return to the definition and formula.

For your notes, label formulas by topic rather than collecting them under “epsilon.” For instance, write “vacuum permittivity: Coulomb’s law” and “engineering strain: change in length over original length.” A context-labeled formula sheet is easier to use when you work on mixed-topic problems.

Frequently asked questions about epsilon

Epsilon can look like one symbol with one meaning, but its definition changes across physics topics. These answers summarize the main distinctions in this guide and give you a quick way to choose the right interpretation before you calculate.

Does epsilon have one universal formula in physics?
No. Its formula depends on the topic and the equation. Check the definition, subscript, and units before using a value.

What is the value of (\varepsilon_0)?
Vacuum permittivity is approximately (8.854\times10^{-12}\ \mathrm{F\,m^{-1}}). Use the precision given in your problem or required by your course.

What is the difference between (\varepsilon) and (\varepsilon_0)?
In electromagnetism, (\varepsilon_0) is vacuum permittivity. An un-subscripted (\varepsilon) may mean a material’s permittivity, but confirm how the equation defines it.

What does (\varepsilon_r) mean?
It is relative permittivity: (\varepsilon_r=\varepsilon/\varepsilon_0). It is dimensionless because it compares quantities with the same units.

Can I use (\varepsilon_r) in place of (\varepsilon_0)?
Not directly. They have different meanings and units. If you need a material’s permittivity, use (\varepsilon=\varepsilon_r\varepsilon_0).

Why is strain written as epsilon?
Engineering strain is commonly defined as the change in length divided by the original length. The ratio is dimensionless, and epsilon is a notation used for it.

What does (|\varepsilon|\ll1) mean?
It means epsilon’s magnitude is much less than one for the approximation being used. The suitable size depends on the required accuracy and the omitted terms.

How can I tell which epsilon a formula uses?
Read its topic and surrounding definitions, check any subscript, and inspect the units. If those clues do not settle it, consult the course text or instructor.

Should strain be written as a decimal or percent?
Either form can be used if you label it clearly. A strain of (0.01) is the same as (1\%); do not treat those numbers as equal without converting.

What is the fastest way to avoid an epsilon error?
Write the symbol’s meaning and units beside the equation before substituting values. Then check that the final units fit the quantity you are calculating.

(This article was written by one of our staff writers, Alan Westbrook. Visit our Meet the Team page.)