[NOTES/EM-01009] From Action at a Distance to Field Concept

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The concept of electric field to describe the interaction of charged particles is briefly explained. Instead of the Coulomb interaction, which is an instantaneous action, we say a charge produces electric field which in turn acts on another charge  present at some other point. Similarly the magnetic force on a current, or a moving charged particle is described in terms of magnetic field produced by a magnet or a current..


Coulomb Force is action at a distance interaction
The Coulomb's force \(\vec F_{12}\) exerted by a charge \(q_1\) at \(\vec r_1\) on another charge \(q_2\) located at \(\vec r_2\) is given by \begin{equation} \vec F_{12} = \frac{q_1q_2}{4\pi\epsilon_0} \frac{\vec r_2-\vec r_1}{|\vec r_2-\vec r_1|^3}. \end{equation} where \(\epsilon_0\) is known as the vacuum permittivity and in SI units it has the value \begin{equation} \epsilon_0 = 8.854\times 10^{-12} {C^2/(Nm^2)}. \end{equation} The Coulomb force between two charges is an action at a distance type force between two charges. The force concept has been replaced with fields and the interaction between two charges is visualized as being a result of local action on a charge due to the electric field created by another charge.

However, it may be added that in the modern version, the interaction between charges arises due to exchange of the photons, the quanta of electromagnetic fields.

The Electric field
For our present purpose of classical electromagnetic theory, we accept the view that a charge \(q_1\) at \(\vec r_1\) produces electric field \(\vec E\) at all points of the space around the charge. This field at a point \(\vec r_2\) is given by \begin{equation}\label{EQ01} \vec E(\vec r_2) = \frac{\vec F_{12}}{q_2} = \frac{q_1}{4\pi\epsilon_0} \frac{\vec r_2-\vec r_1}{|\vec r_2-\vec r_1|^3}. \end{equation} The force felt by a charge \(q_2\) located at \(\vec r_2\), is then \(q_2 \vec E (\vec r_2)\), same as the Coulomb force expression \eqRef{EQ01}. The constant \(\epsilon_0\) is called {\it vacuum permittivity}, or the permittivity of free space. If we have several charges \(q_k, k=1,..N\), their force on a charge \(q\) will be a sum of forces due to the individual charges \(q_k\). Translated in terms of electric field, this statement implies the {\bf superposition principle} for the electric field. The electric field due to a system of several charges \(q_k, k=1,..N\), is just the sum of electric fields due to individual charges and is given by \begin{equation}\label{eps} \vec E (\vec r) =\frac{1}{4\pi\epsilon_0} \sum _{k=1}^N\frac{q_k(\vec r-\vec r_k)}{|\vec r-\vec r_k|^3}. \end{equation}

Magnetic Field
Just as two charges exert forces on each other, a current carrying conductor exerts a magnetic force on another current carrying conductor. For example, two thin, long, parallel straight wires carrying currents \(I_1\), \(I_2\) and separated by a distance \(d\) attract each other by a force per unit length \begin{equation}\label{mu} F = \frac{\mu_0}{4\pi}\frac{I_1 I_2}{d}. \end{equation} The constant \(\mu_0\) appearing here is called the {\it vacuum permeability} or the permeability of the free space. It has numerical value \begin{equation} \mu_0=(4\pi) \times 10^{-7}(N/A^2) \end{equation} Apart from the magnetic force between currents,  a magnet also exerts a force on a current carrying conductor.

The magnetic force on a current is visualized as being due to magnetic field produced by a current, or a magnet. A current consists of flow of charges, thus we have magnetic force on moving charges.

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