Coulomb's Law Information

Coulomb's law, or Coulomb's inverse square law, is a law of physics describing the electrostatic interaction between electrically charged particles. It was studied and first published in 1783 by French physicist Charles Augustin de Coulomb and was essential to the development of the theory of electromagnetism. Nevertheless, the dependence of the electric force with distance had been proposed previously by Joseph Priestley^[1] and the dependence with both distance and charge had been discovered, but not published, by Henry Cavendish, prior to Coulomb's works.

Basic Equation

It states that : "The magnitude of the Electrostatics force of interaction between two point charges is directly proportional to the scalar multiplication of the magnitudes of charges and inversely proportional to the square of the distances between them."

The scalar form of Coulomb's law is an expression for the magnitude and sign of the electrostatic force between two idealized point charges, small in size compared to their separation. This force (F) acting simultaneously on point charges (q₁) and (q₂), is given by

where r is the separation distance and k_e is a proportionality constant. A positive force implies it is repulsive, while a negative force implies it is attractive.^[2] The proportionality constant k_e, called the Coulomb constant (sometimes called the Coulomb force constant), is related to defined properties of space and can be calculated based on knowledge of empirical measurements of the speed of light:^[3]

In SI units, the meter is defined such that the speed of light in vacuum (or electromagnetic waves, in general), denoted c,^[4] is exactly 299,792,458 m·s^−1[5], and the magnetic constant (μ₀) is set at 4π × 10⁻⁷ H·m⁻¹.^[6] In agreement with electromagnetic theory, requiring that

the value for the electric constant (ε₀) is derived to be ε₀ = 1/(μ₀c²) ≈ 8.85418782×10−12 F·m⁻¹.^[7] In cgs units, the unit charge, esu or statcoulomb, is defined in such a way that the Coulomb constant is 1 and dimensionless.

In the more useful vector-form statement, the force in the equation is a vector force acting on either point charge, so directed as to push it away from the other point charge; the right-hand side of the equation, in this case, must have an additional product term of a unit vector pointing in one of two opposite directions, e.g., from q₁ to q₂ if the force is acting on q₂; the charges may have either sign and the sign of their product determines the ultimate direction of that force. Thus, the vector force pushing the charges away from each other (pulling towards each other if negative) is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. The square of the distance part arises from the fact that the force field due to an isolated point charge is uniform in all directions and gets "diluted" with distance as much as the area of a sphere centered on the point charge expands with its radius.

The law of superposition allows this law to be extended to include any number of point charges, to derive the force on any one point charge by a vector addition of these individual forces acting alone on that point charge. The resulting vector happens to be parallel to the electric field vector at that point, with that point charge (or "test charge") removed.

Coulomb's law can also be interpreted in terms of atomic units with the force expressed in Hartrees per Bohr radius, the charge in terms of the elementary charge, and the distances in terms of the Bohr radius.

Electric field

It follows from the Coulomb's Law that the magnitude of the electric field (E) created by a single point charge (q) at a certain distance (r) is given by:

For a positive charge, the direction of the electric field points along lines directed radially away from the location of the point charge, while the direction is the opposite for a negative charge. The SI units of electric field are volts per metre or newtons per coulomb.

Vector form

In order to obtain both the magnitude and direction of the force on a charge, q₁ at position , experiencing a field due to the presence of another charge, q₂ at position , the full vector form of Coulomb's law is required.

where r is the separation of the two charges. This is simply the scalar definition of Coulomb's law with the direction given by the unit vector, , parallel with the line from charge q₂ to charge q₁.^[8]

If both charges have the same sign (like charges) then the product q₁q₂ is positive and the direction of the force on q₁ is given by ; the charges repel each other. If the charges have opposite signs then the product q₁q₂ is negative and the direction of the force on q₁ is given by ; the charges attract each other.

System of discrete charges

The principle of linear superposition may be used to calculate the force on a small test charge, q, due to a system of N discrete charges:

where q_i and are the magnitude and position respectively of the i^th charge, is a unit vector in the direction of (a vector pointing from charge q_i to charge q), and R_i is the magnitude of (the separation between charges q_i and q).^[8]

Continuous charge distribution

For a charge distribution an integral over the region containing the charge is equivalent to an infinite summation, treating each infinitesimal element of space as a point charge dq.

For a linear charge distribution (a good approximation for charge in a wire) where gives the charge per unit length at position , and is an infinitesimal element of length,

.^[9]

For a surface charge distribution (a good approximation for charge on a plate in a parallel plate capacitor) where gives the charge per unit area at position , and is an infinitesimal element of area,

For a volume charge distribution (such as charge within a bulk metal) where gives the charge per unit volume at position , and is an infinitesimal element of volume,

^[8]

The force on a small test charge at position is given by

Graphical representation

Below is a graphical representation of Coulomb's law, when q₁q₂ > 0. The vector is the force experienced by q₁. The vector is the force experienced by q₂. Their magnitudes will always be equal. The vector is the displacement vector between two charges (q₁ and q₂).

Electrostatic approximation

In either formulation, Coulomb’s law is fully accurate only when the objects are stationary, and remains approximately correct only for slow movement. These conditions are collectively known as the electrostatic approximation. When movement takes place, magnetic fields are produced which alter the force on the two objects. The magnetic interaction between moving charges may be thought of as a manifestation of the force from the electrostatic field but with Einstein’s theory of relativity taken into consideration.

Table of derived quantities

See also

• Biot–Savart law
• Method of image charges
• Electric field
• Electric constant
• Coulomb, the SI unit of electric charge named after Charles Augustin de Coulomb
• Electromagnetic force
• Molecular modelling
• Static forces and virtual-particle exchange
• Darwin Lagrangian

Notes

1. ^ Robert S. Elliott (1999). Electromagnetics: History, Theory, and Applications. ISBN 978-0-7803-5384-8. http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0780353846.html
2. ^ Coulomb's law, Hyperphysics
3. ^ Coulomb's constant, Hyperphysics
4. ^ Current practice is to use c₀ to denote the speed of light in vacuum according to ISO 31. In the original Recommendation of 1983, the symbol c was used for this purpose and continues to be commonly used. See NIST Special Publication 330, Appendix 2, p. 45
5. ^ Base unit definitions: Meter. Physics.nist.gov. Retrieved on 2010-09-28.
6. ^ Base unit definitions: Ampere. Physics.nist.gov. Retrieved on 2010-09-28.
7. ^ CODATA Value: electric constant. Physics.nist.gov. Retrieved on 2010-09-28.
8. ^ ^{a b c} Coulomb's law, University of Texas
9. ^ Charged rods, PhysicsLab.org

References

• Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. ISBN 0-13-805326-X.
• Tipler, Paul (2004). Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.). W. H. Freeman. ISBN 0-7167-0810-8.

External links

• Coulomb's Law on Project PHYSNET.
• Electricity and the Atom — a chapter from an online textbook
• A maze game for teaching Coulomb's Law—a game created by the Molecular Workbench software
• The inverse cube law The inverse cube law for dipoles (PDF file) by Eng. Xavier Borg
• Electric Charges, Polarization, Electric Force, Coulomb's Law Walter Lewin, 8.02 Electricity and Magnetism, Spring 2002: Lecture 1 (video). MIT OpenCourseWare. License: Creative Commons Attribution-Noncommercial-Share Alike.

Categories: Electrostatics | Introductory physics | Fundamental physics concepts

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