Electrostatics
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Electrostatics is a branch of physics that studies slow-moving or stationary electric charges.
Since
There are many examples of electrostatic phenomena, from those as simple as the attraction of plastic wrap to one's hand after it is removed from a package, to the apparently spontaneous explosion of grain silos, the damage of electronic components during manufacturing, and photocopier and laser printer operation.
The electrostatic model accurately predicts electrical phenomena in "classical" cases where the velocities are low and the system is macroscopic so no quantum effects are involved. It also plays a role in quantum mechanics, where additional terms also need to be included.
Coulomb's law
Coulomb's law states that:[5]
'The magnitude of the electrostatic force of attraction or repulsion between two point charges is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them.'
The force is along the straight line joining them. If the two charges have the same sign, the electrostatic force between them is repulsive; if they have different signs, the force between them is attractive.
If is the distance (in
where ε0 is the vacuum permittivity, or permittivity of free space:[6]
The
These physical constants (ε0, ke, e) are currently defined so that e is exactly defined, and ε0 and ke are measured quantities.
Electric field
The electric field, , in units of Newtons per Coulomb or volts per meter, is a vector field that can be defined everywhere, except at the location of point charges (where it diverges to infinity).[8] It is defined as the electrostatic force in newtons on a hypothetical small
Electric field lines are useful for visualizing the electric field. Field lines begin on positive charge and terminate on negative charge. They are parallel to the direction of the electric field at each point, and the density of these field lines is a measure of the magnitude of the electric field at any given point.
Consider a collection of particles of charge , located at points (called source points), the electric field at (called the field point) is:[8]
where is the
Gauss' law
Gauss's law[9][10] states that "the total electric flux through any closed surface in free space of any shape drawn in an electric field is proportional to the total electric charge enclosed by the surface." Many numerical problems can be solved by considering a Gaussian surface around a body. Mathematically, Gauss's law takes the form of an integral equation:
where is a volume element. If the charge is distributed over a surface or along a line, replace by or . The divergence theorem allows Gauss's Law to be written in differential form:
where is the divergence operator.
Poisson and Laplace equations
The definition of electrostatic potential, combined with the differential form of Gauss's law (above), provides a relationship between the potential Φ and the charge density ρ:
This relationship is a form of Poisson's equation.[11] In the absence of unpaired electric charge, the equation becomes Laplace's equation:
Electrostatic approximation
The validity of the electrostatic approximation rests on the assumption that the electric field is
From Faraday's law, this assumption implies the absence or near-absence of time-varying magnetic fields:
In other words, electrostatics does not require the absence of magnetic fields or electric currents. Rather, if magnetic fields or electric currents do exist, they must not change with time, or in the worst-case, they must change with time only very slowly. In some problems, both electrostatics and magnetostatics may be required for accurate predictions, but the coupling between the two can still be ignored. Electrostatics and magnetostatics can both be seen as non-relativistic Galilean limits for electromagnetism.[12] In addition, conventional electrostatics ignore quantum effects which have to be added for a complete description.[8]: 2
Electrostatic potential
As the electric field is
The gradient theorem can be used to establish that the electrostatic potential is the amount of work per unit charge required to move a charge from point to point with the following line integral:
From these equations, we see that the electric potential is constant in any region for which the electric field vanishes (such as occurs inside a conducting object).
Electrostatic energy
A test particle's potential energy, , can be calculated from a line integral of the work, . We integrate from a point at infinity, and assume a collection of particles of charge , are already situated at the points . This potential energy (in Joules) is:
where is the distance of each charge from the
where the following sum from, j = 1 to N, excludes i = j:
This electric potential, is what would be measured at if the charge were missing. This formula obviously excludes the (infinite) energy that would be required to assemble each point charge from a disperse cloud of charge. The sum over charges can be converted into an integral over charge density using the prescription :
This second expression for
Electrostatic pressure
On a conductor, a surface charge will experience a force in the presence of an electric field. This force is the average of the discontinuous electric field at the surface charge. This average in terms of the field just outside the surface amounts to:
This pressure tends to draw the conductor into the field, regardless of the sign of the surface charge.
See also
- Electromagnetism – Fundamental interaction between charged particles
- Electrostatic generator, machines that create static electricity.
- Electrostatic induction, separation of charges due to electric fields.
- Permittivity and relative permittivity, the electric polarizability of materials.
- Quantisation of charge, the charge units carried by electrons or protons.
- Static electricity, stationary charge accumulated on a material.
- Triboelectric effect, separation of charges due to sliding or contact.
References
- ISBN 9781947172210. Ch.30: Conductors, Insulators, and Charging by Induction
- ISBN 9781119013846.
- ^ "Polarization". Static Electricity - Lesson 1 - Basic Terminology and Concepts. The Physics Classroom. 2020. Retrieved 18 June 2021.
- ^ Thompson, Xochitl Zamora (2004). "Charge It! All About Electrical Attraction and Repulsion". Teach Engineering: Stem curriculum for K-12. University of Colorado. Retrieved 18 June 2021.
- ISBN 978-1-108-33351-1. Retrieved 2023-08-11.
- ISBN 9780195387759.
- ^ "SI Units". NIST. 2010-04-12.
- ^ ISBN 978-1107014022.
- . Retrieved 2023-08-11.
- ISBN 978-3-642-49320-1, retrieved 2023-08-11
- ^ Poisson, M; sciences (France), Académie royale des (1827). Mémoires de l'Académie (royale) des sciences de l'Institut (imperial) de France. Vol. 6. Paris.
- S2CID 118443242.
Further reading
- Hermann A. Haus; James R. Melcher (1989). Electromagnetic Fields and Energy. Englewood Cliffs, NJ: Prentice-Hall. ISBN 0-13-249020-X.
- Halliday, David; Robert Resnick; Kenneth S. Krane (1992). Physics. New York: John Wiley & Sons. ISBN 0-471-80457-6.
- ISBN 0-13-805326-X.
External links
- Media related to Electrostatics at Wikimedia Commons
- The Feynman Lectures on Physics Vol. II Ch. 4: Electrostatics
- Introduction to Electrostatics: Point charges can be treated as a distribution using the Dirac delta function
Learning materials related to Electrostatics at Wikiversity