Electric Field of a Point Charge

In one sentence: The electric field around a charge tells you how strongly, and in which direction, a small charged object placed nearby would be pushed or pulled.

$E = \frac{k_e\, q}{r^2}$

Read it out loud: “E equals k sub e times q, divided by r squared. The electric field E equals the Coulomb constant, written k sub e, times the source charge q, divided by the distance r squared.”

Before you start

The ideas and signs this page uses, explained from zero. For powers of ten, Greek letters and other notation in general, see Math basics.

Electric charge and the coulomb (C): Electric charge is a property of matter that makes it push or pull other charged things. There are two kinds, positive and negative. Like charges repel, and opposite charges attract. The unit is the coulomb (C). One coulomb is a very large amount of charge: it equals the charge of about 6.24 billion billion electrons, which is $6.24 \times 10^{18}$ electrons.
Powers of ten, like $10^{9}$ and $10^{-6}$: A power of ten is ten multiplied by itself a set number of times. $10^{9}$ is a 1 followed by nine zeros, which is one billion. A negative power means divide: $10^{-6}$ is one millionth, 0.000001. In $k_e = 8.9875517923 \times 10^{9}$ , the power moves the decimal point nine places to the right, giving 8,987,551,792.3. The prefix micro (µ) means $10^{-6}$ , so $2\ \mu\text{C} = 2 \times 10^{-6}$ C.
Force and the newton (N): A force is a push or a pull. Its unit is the newton (N), which is the force that gives a 1 kg mass an acceleration of 1 meter per second every second. The weight of a 100 gram apple is about 1 N, because Earth’s gravity pulls on each kilogram with about 9.8 N.
Squares, like $r^2$: Writing $r^2$ means $r \times r$ , the square of $r$ . If $r = 0.5$ m, then $r^2 = 0.25$ square meters. Doubling $r$ does not double $r^2$ . It multiplies it by four, because $2r \times 2r = 4r^2$ . The field divides by $r^2$ , so doubling the distance cuts the field to one quarter.
Vectors and direction: Some quantities have only a size, like a mass or a temperature. A vector has both a size and a direction, like wind, which blows at a certain speed toward a certain side. The electric field is a vector. At each point it has a strength, in newtons per coulomb, and a direction, which is shown by an arrow.
Test charge $q_0$: A test charge is a small charge placed at a point to measure the field there. It is only a measuring tool and is not part of the field. The field is the force on the test charge divided by its charge, so the field does not depend on how big the test charge is.

What each symbol means

Symbol	Meaning
$E$	Electric field, the force per unit charge at a point, measured in newtons per coulomb (N/C). One newton per coulomb equals one volt per meter. A balloon rubbed on hair can pull small bits of paper toward it, because it sets up an electric field around itself.
$k_e$	The Coulomb constant, $k_e = 8.9875517923 \times 10^{9}$ N m $^2$ /C $^2$ . It is a fixed number that sets how strong electric forces are in the SI units of meters, newtons and coulombs. Its units cancel with the charge and distance units, so the field comes out in newtons per coulomb.
$q$	The source charge, the charge that creates the field, measured in coulombs (C). Its size sets the strength of the field and its sign sets the direction. In the example it is 2 microcoulombs, which is $2 \times 10^{-6}$ C.
$r$	The distance from the source charge to the point where the field is measured, in meters (m). Because the distance is squared, a point twice as far away feels one quarter of the field. In the example $r$ is 0.5 m, which is half a meter.

The formula piece by piece

$r$: Start with the distance $r$ , in meters, from the charge to the point. Everything else in the formula depends on how far away the point is, and the distance is the only thing that describes the position of the point.
$r^{2}$: Square the distance by multiplying it by itself. The square comes from the surface of a sphere of radius $r$ , whose area is $4\pi r^2$ . The field lines leaving the charge spread over a larger sphere as the distance grows, so the field gets weaker with the square of the distance.
$q$: The source charge, with its sign. It is the amount of charge that makes the field, in coulombs. The bigger the charge, the stronger the field, in direct proportion, and the sign decides which way the field arrows point.
$k_e\, q$: Multiply the Coulomb constant by the charge. This product is the strength of the source before the distance is taken into account. Its units are newton meters squared per coulomb, and dividing by the distance squared finishes the job.
$E = \frac{k_e\, q}{r^2}$: The whole formula in one sentence: the electric field at a point equals the Coulomb constant times the source charge, divided by the square of the distance from the charge to the point.
$F = q_0 E$: The force on a charge $q_0$ placed in a field $E$ . Multiply the field by the charge to get the force in newtons. This is how the field is used in practice: once you know $E$ at a point, the force on any charge placed there is $q_0 E$ .

What the formula is doing, and why

A point charge creates an electric field in the space around it. The field at a point is the force a small test charge would feel there, divided by the size of that test charge: $E = F / q_0$ . Because the test charge is divided out, the field is a property of the space itself, not of the test charge. Its unit is newtons per coulomb (N/C).

Why divide by $r^2$ ? Imagine a sphere of radius $r$ around the charge. Its surface area is $4\pi r^2$ . The same field lines leaving the charge cross this surface, and a bigger sphere has more area for them to cover. So at a larger distance the field is spread over more area, and the field at each spot is weaker, by a factor of $r^2$ .

Why multiply by $q$ and $k_e$ ? Double the source charge and twice as many field lines leave it, so the field doubles. The Coulomb constant $k_e$ does not change the shape of the law. It sets the scale, so that the answer comes out in newtons per coulomb when the distance is in meters and the charge is in coulombs.

Which way does it point? The sign of $q$ sets the direction. A positive source pushes a positive test charge away, so the field arrows point outward. A negative source pulls a positive test charge in, so the arrows point inward. A small positive test charge placed in the field is pushed along the arrow at its spot.

When it applies. The formula is exact for one point charge in empty space. For a charged sphere, the same formula gives the field outside the sphere, as if all of its charge sat at the center. When several charges are present, find the field of each one separately and add the results as vectors, arrow by arrow. The formula does not add them for you.

An everyday picture

Think of a wind map over a city. At every spot there is a wind speed and a direction, even if nobody is standing there. The electric field is the same kind of map: at each point it says how hard, and in which direction, a charge placed there would be pushed. Near a strong source the arrows are long, and farther away they shrink quickly, because the field falls off with the square of the distance.

Worked example: Field 0.5 m from a 2 microcoulomb charge

A point charge of $q = 2\ \mu\text{C}$ , which is $2 \times 10^{-6}$ C, sits in empty space. Find the electric field at a point 0.5 m away. Then find the force on a 3 microcoulomb charge placed at that point.

Convert the charge to coulombs. The prefix micro, written $\mu$ , means one millionth, which is $10^{-6}$ . So $2\ \mu\text{C} = 2 \times 10^{-6}$ C. Multiplying by $10^{-6}$ moves the decimal point six places to the left, from 2 to 0.000002 coulombs.
Square the distance. $r^2 = (0.5\ \text{m})^2 = 0.5 \times 0.5 = 0.25\ \text{m}^2$ . The unit is square meters, because a meter times a meter is a square meter of area.
Multiply the constant by the charge. $k_e q = 8.9875517923 \times 10^{9} \times 2 \times 10^{-6}$ . Multiply the ordinary numbers first, $8.9875517923 \times 2 = 17.975$ . Then add the exponents, $9 + (-6) = 3$ , so the powers of ten give $10^{3}$ . The product is $17.975 \times 10^{3} = \text{17,975}$ , in units of newton meters squared per coulomb.
Divide by the squared distance. $E = \frac{\text{17,975}}{0.25} \approx \text{71,900}$ N/C. Dividing by 0.25 is the same as multiplying by 4, because 0.25 is one quarter. In scientific notation this is $7.19 \times 10^{4}$ N/C, since the decimal point moves four places to the left.
Find the force on the 3 microcoulomb charge. The field gives the force per coulomb, so the force on a charge is $F = qE$ . Here $q = 3 \times 10^{-6}$ C and $E = 7.19 \times 10^{4}$ N/C. The exponents add, $-6 + 4 = -2$ , so $3 \times 7.19 = 21.57$ with $10^{-2}$ , which gives $F \approx 0.216$ N.
Check the units. Newtons per coulomb times coulombs gives newtons, because the coulombs cancel. So the answer is a force, as it should be, and it is measured in newtons.
Say the force in everyday terms. A force of 0.216 N is about the weight of a 22 gram object. Weight is the force of gravity, about 9.8 N for each kilogram, so 0.216 divided by 9.8 is about 0.022 kg, which is 22 grams.

Answer: The field at 0.5 m is about $7.19 \times 10^{4}$ N/C, pointing away from the positive charge. A 3 microcoulomb charge placed there feels a force of about 0.216 N, pushing it away from the source. That force is about the weight of a 22 gram object.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Using microcoulombs as if they were coulombs. Write 2 microcoulombs as $2 \times 10^{-6}$ C. If you skip the $10^{-6}$ , the field comes out a million times too big.
Forgetting that the field falls off with $r^2$ . Doubling the distance cuts the field to one quarter, not one half.
Mixing up the field and the force. The field is the force per unit charge, so the force on a charge is $F = qE$ . The field is measured in N/C and the force in newtons.
Giving the field a direction from its size alone. The sign of the source charge sets the direction, so a positive source points its field outward and a negative source points it inward.

Where you will meet it

Designing high-voltage equipment, where the field strength decides when air breaks down and a spark jumps.
Electron microscopes and particle accelerators, which use electric fields to steer and speed up charged particles.
Electrostatic precipitators in smokestacks, which charge dust particles and pull them out of the smoke.
Static electricity, such as a charged balloon pulling paper bits toward it or a small shock from a doorknob.

Questions people ask

Why is the field measured in N/C?

The field is a force divided by a charge. Newtons divided by coulombs gives newtons per coulomb, and $1\ \text{N/C} = 1\ \text{V/m}$ . Both units describe the same field, one as force per charge and the other as voltage per meter.

Does the field depend on the test charge?

No. The test charge is only used to measure the field. At a given point the field is set by the source charge and the distance, so it is a property of the space itself.

Which way does the field point near a negative charge?

Toward the charge. A positive test charge is pulled in, so the field arrows point inward.

What happens to the field if the distance doubles?

It falls to one quarter of its old value. For the 2 microcoulomb charge, the field at 0.5 m is about 71,900 N/C, and at 1.0 m it is about 17,975 N/C.

Coulomb's Law: Coulomb’s law tells you how hard two electric charges push or pull each other. The force gets weaker fast as they move apart, because it drops off with the square of the distance.
Electric Potential: Electric potential tells you how much energy each coulomb of charge would carry at one spot in space, and it is measured in volts.
Charge and Current: Electric current is how much charge passes a point each second. The same charge moving in less time means a bigger current.
Parallel-Plate Capacitor: The parallel-plate formula says a capacitor holds more charge per volt when its plates are bigger, closer together, or separated by a material with a higher permittivity.