Force between two electric charges. The magnitude of the force math.F is proportional to the product of the two charges, math.q1 and math.q2, divided by the square of the distance math.r between them, or math.F = math.kmath.q1math.q2/math.r2, where math.k is a constant that depends on the measurement system being used. The Coulomb force can be one of repulsion, such as the force between two objects having like charges, or it can be attractive, such as the force between two objects having opposite charges.
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If one does not require the specific direction of the force then the simplified, scalar, version of Coulomb's law will suffice. The magnitude of the force on a charge, , due to the presence of a second charge, , is given by the magnitude of
where is the separation of the charges and is the electric constant. A positive force implies a repulsive interaction, while a negative force implies an attractive interaction.
The prefactor, termed the Coulomb's constant (), is:
In SI units the speed of light in vacuum c0 is defined as the numerical value c0 = 299 792 458 m s−1 (See c0) and the magnetic constant μ0 is defined as 4π x 10−7 H · m−1 (See μ0), leading to the definition for the electric constant of ε0 = 1/(μ0c02) ≈ 8.854 187 817 x 10−12 F m−1 (See NIST ε0) In cgs units, the unit charge, esu of charge or statcoulomb, is defined so that this Coulomb force constant is 1.
This formula says that the magnitude of the force is directly proportional to the magnitude of the charges of each object and inversely proportional to the square of the distance between them. The exponent in Coulomb's Law has been found to differ from −2 by less than one in a billion.
When measured in units that people commonly use (such as SI—see International System of Units), the electrostatic force constant, , is numerically much much larger than the universal gravitational constant . This means that for objects with charge that is of the order of a unit charge (C) and mass of the order of a unit mass (kg), the electrostatic forces will be so much larger than the gravitational forces that the latter force can be ignored. This is not the case when Planck units are used and both charge and mass are of the order of the unit charge and unit mass. However, charged elementary particles have mass that is far less than the Planck mass while their charge is about the Planck charge so that, again, gravitational forces can be ignored. For example, the electrostatic force between an electron and a proton, which constitute a hydrogen atom, is almost 40 orders of magnitude greater than the gravitational force between them.
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.
For a positive charge , the direction of points along lines directed radially away from the location of the point charge, while the direction is the opposite for a negative charge. The units of electric field are volts per meter or newtons per coulomb.
In order to obtain both the magnitude and direction of the force on a charge, at position , experiencing a field due to the presence of another charge, at position , the full vector form of Coulomb's law is required.
where is the separation of the two charges. Note that this is simply the scalar definition of Coulomb's law with the direction given by the unit vector, , parallel with the line from charge to charge .
If both charges have the same sign (like charges) then the product is positive and the direction of the force on is given by ; the charges repel each other. If the charges have opposite signs then the product is negative and the direction of the force on is given by ; the charges attract each other.
The principle of linear superposition may be used to calculate the force on a small test charge, , due to a system of discrete charges:
where and are the magnitude and position respectively of the charge, is a unit vector in the direction of (a vector pointing from charge to charge ), and is the magnitude of (the separation between charges and ).
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,
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,
The force on a small test charge at position is given by
Below is a graphical representation of Coulomb's law, when . The vector is the force experienced by . The vector is the force experienced by . Their magnitudes will always be equal. The vector is the displacement vector between two charges ( and ).
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.
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