For small distances and low field strengths, such interactions are better described by quantum electrodynamics. Fundamental physical aspects of classical electrodynamics are presented in many texts, such as those by Feynman , Leighton and Sands ,  Griffiths ,  Panofsky and Phillips,  and Jackson.
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The physical phenomena that electromagnetism describes have been studied as separate fields since antiquity. For example, there were many advances in the field of optics centuries before light was understood to be an electromagnetic wave. However, the theory of electromagnetism , as it is currently understood, grew out of Michael Faraday 's experiments suggesting an electromagnetic field and James Clerk Maxwell 's use of differential equations to describe it in his A Treatise on Electricity and Magnetism For a detailed historical account, consult Pauli,  Whittaker,  Pais,  and Hunt.
The electromagnetic field exerts the following force often called the Lorentz force on charged particles:.
Electromagnetism - Wikipedia
The above equation illustrates that the Lorentz force is the sum of two vectors. One is the cross product of the velocity and magnetic field vectors. Based on the properties of the cross product, this produces a vector that is perpendicular to both the velocity and magnetic field vectors.
The other vector is in the same direction as the electric field. The sum of these two vectors is the Lorentz force.
Classical Electrodynamics, 3rd Edition
Therefore, in the absence of a magnetic field, the force is in the direction of the electric field, and the magnitude of the force is dependent on the value of the charge and the intensity of the electric field. In the absence of an electric field, the force is perpendicular to the velocity of the particle and the direction of the magnetic field. If both electric and magnetic fields are present, the Lorentz force is the sum of both of these vectors. The electric field E is defined such that, on a stationary charge:.
The size of the charge doesn't really matter, as long as it is small enough not to influence the electric field by its mere presence.
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In electrostatics, where charges are not moving, around a distribution of point charges, the forces determined from Coulomb's law may be summed. The result after dividing by q 0 is:.
If the field is instead produced by a continuous distribution of charge, the summation becomes an integral:. Both of the above equations are cumbersome, especially if one wants to determine E as a function of position. A scalar function called the electric potential can help.
Electric potential, also called voltage the units for which are the volt , is defined by the line integral. Unfortunately, this definition has a caveat. As a result, one must add a correction factor, which is generally done by subtracting the time derivative of the A vector potential described below.
Suitable for first-year graduate students in physics who have taken an undergraduate course in electromagnetism, it focuses on core concepts and related aspects of math and physics. Each stage of the theory is carefully developed in a clear and systematic approach that integrates mathematics and physics so that readers are introduced to the theory and learn the mathematical skills in context of real physics applications. Topics include methods of solution in electrostatics, Green's functions, electrostatics in matter, magnetism and ferromagnetism, electromagnetic waves in matter, special relativity, and the electrodynamics of moving bodies.
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