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18.4 Coulomb’s law

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Learning objectives

By the end of this section, you will be able to:

  • State Coulomb's law in terms of how the electrostatic force changes with the distance between two objects.
  • Calculate the electrostatic force between two point charges, such as electrons or protons.
  • Compare the electrostatic force to the gravitational attraction for a proton and an electron; for a human and the Earth.

The information presented in this section supports the following AP® learning objectives and science practices:

  • 3.A.3.3 The student is able to describe a force as an interaction between two objects and identify both objects for any force. (S.P. 1.4)
  • 3.A.3.4 The student is able to make claims about the force on an object due to the presence of other objects with the same property: mass, electric charge. (S.P. 6.1, 6.4)
  • 3.C.2.1 The student is able to use Coulomb's law qualitatively and quantitatively to make predictions about the interaction between two electric point charges (interactions between collections of electric point charges are not covered in Physics 1 and instead are restricted to Physics 2). (S.P. 2.2, 6.4)
  • 3.C.2.2 The student is able to connect the concepts of gravitational force and electric force to compare similarities and differences between the forces. (S.P. 7.2)
Two spiral galaxies show the strong gravitational attraction between them as their arms appear to reach out toward one another.
This NASA image of Arp 87 shows the result of a strong gravitational attraction between two galaxies. In contrast, at the subatomic level, the electrostatic attraction between two objects, such as an electron and a proton, is far greater than their mutual attraction due to gravity. (credit: NASA/HST)

Through the work of scientists in the late 18th century, the main features of the electrostatic force    —the existence of two types of charge, the observation that like charges repel, unlike charges attract, and the decrease of force with distance—were eventually refined, and expressed as a mathematical formula. The mathematical formula for the electrostatic force is called Coulomb's law    after the French physicist Charles Coulomb (1736–1806), who performed experiments and first proposed a formula to calculate it.

Coulomb's law

F = k | q 1 q 2 | r 2 . size 12{F=k { {q rSub { size 8{1} } q rSub { size 8{2} } } over {r rSup { size 8{2} } } } } {}

Coulomb's law calculates the magnitude of the force F between two point charges, q 1 size 12{q rSub { size 8{1} } } {} and q 2 size 12{q rSub { size 8{2} } } {} , separated by a distance r . In SI units, the constant k is equal to

k = 8 . 988 × 10 9 N ⋅ m 2 C 2 ≈ 8 . 99 × 10 9 N ⋅ m 2 C 2 . size 12{k=8 "." "988" times "10" rSup { size 8{9} } { {N cdot m rSup { size 8{2} } } over {C rSup { size 8{2} } } } approx 9 "." "00" times "10" rSup { size 8{9} } { {N cdot m rSup { size 8{2} } } over {C rSup { size 8{2} } } } } {}

The electrostatic force is a vector quantity and is expressed in units of newtons. The force is understood to be along the line joining the two charges. (See [link] .)

Although the formula for Coulomb's law is simple, it was no mean task to prove it. The experiments Coulomb did, with the primitive equipment then available, were difficult. Modern experiments have verified Coulomb's law to great precision. For example, it has been shown that the force is inversely proportional to distance between two objects squared F ∝ 1 / r 2 size 12{ left (F prop {1} slash {r rSup { size 8{2} } } right )} {} to an accuracy of 1 part in 10 16 size 12{"10" rSup { size 8{"16"} } } {} . No exceptions have ever been found, even at the small distances within the atom.

In part a, two charges q one and q two are shown at a distance r. Force vector arrow F one two points toward left and acts on q one. Force vector arrow F two one points toward right and acts on q two. Both forces act in opposite directions and are represented by arrows of same length. In part b, two charges q one and q two are shown at a distance r. Force vector arrow F one two points toward right and acts on q one. Force vector arrow F two one points toward left and acts on q two. Both forces act toward each other and are represented by arrows of same length.
The magnitude of the electrostatic force F size 12{F} {} between point charges q 1 size 12{q rSub { size 8{1} } } {} and q 2 size 12{q rSub { size 8{2} } } {} separated by a distance r size 12{F} {} is given by Coulomb's law. Note that Newton's third law (every force exerted creates an equal and opposite force) applies as usual—the force on q 1 size 12{q rSub { size 8{1} } } {} is equal in magnitude and opposite in direction to the force it exerts on q 2 size 12{q rSub { size 8{2} } } {} . (a) Like charges. (b) Unlike charges.
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Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
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