[NOTES/EM-02001] -Coulomb’s Law and Electric Field

 Force due to a system of point charges

Consider a system of two charges, \(Q\) at a position $\vec{r}$ and \(q\) at  position $\vec{r}_0$.
Let \(F\) is the force on the charge \(q\) due to charge \(Q\) and   \(\vec{F}^\prime\) be the force on the charge \(Q\) due to the charge \(q\). Then the magnitudes of \(\vec F\) and \(\vec F^\prime\) are equal in magnitude
\begin{equation}\label{EQ02}
|\vec F| = |\vec F^\prime| =\frac{1}{4\pi\epsilon_0}\frac{qQ}{R^2}, \qquad \vec R = \vec r-\vec r_0
\end{equation}
and obey the inverse square law,
where \(R\) is the distance  between the charges.


The forces \(\vec F, \vec F\prime\)  act along the  line joining the two charges and is equal and opposite in the sense of Newton's third law.\\
The force \(\vec{F}, \vec F^\prime\) on the charge \(Q\) due to the charge \(q\) can be written as a vector equation
\begin{equation}\label{EQ01}
 \vec{F} =\frac{qQ}{4\pi\epsilon_0}\frac{(\vec{r}-\vec{r}_0)}{|\vec{r}-\vec{r}_0|^3}
\end{equation}
Here \(\vec r, \vec r_0\) are the positions of charges \(Q,q\) respectively.

The force on a given point charge \(Q\) located at $\vec{r}$ due to several point
charges $q_1,q_2,\ldots, q_N$ is given by the vector sum of the force on \(Q\) due to each
of the charges $q_1,q_2,\ldots, q_N.$
If position of $ q_1$ is  $\vec{r_1}$, $q_2$ is  $\vec{r_2} , \ldots $, then
\begin{eqnarray}\label{EQ04}
\vec{F}=\sum_k\Big(\frac{q_k\,Q}{4\pi\epsilon_0}\Big)\frac{(\vec{r}-\vec{r}_k)}{ \mid\vec{r}-\vec{r}_k\mid^3}.
\end{eqnarray}

We say that the force due to system of point charges is obtained by {\it superposition of forces} due to the individual charges.

Electric field and the superposition principle

Faraday introduced the concept of  electric field. A charge produces {\tt electric field} in the space around it. When there is another charge located at some point \(P\),  the field exerts a force on the  point charge. The field exists even if  no other
charge is present.
The Coulomb force between charges acts instantaneously. If one of the charges is moved a little, change in its force should be felt
immediately
on other charges present even if they are very far off say millions of light years away.
This is inconsistent with special theory of relativity which says that  no signal can travel faster than light.
The concept of electric field
bypasses this problem.

The electric field due to  system of charges is defined as the force on unit positive charge.
Thus the electric field at a point \(\vec{r}\)  due to a  point charge \(q_0\) located at \(\vec{r}_0\) is
\begin{equation}\label{EQ03}
 \vec{E} =\frac{q_0}{4\pi\epsilon_0}\frac{(\vec{r}-\vec{r}_0)}{|\vec{r}-\vec{r}_0|^3}  
 \end{equation} Let $q_1,q_2,q_3,\ldots, q_N$ be a system of \(N\) charges located at positions $\vec{r}_1,\vec{r}_2,\ldots,\vec{r}_N$. Thus the electric field due to these charges at point $\vec{r}$, being the force on  {\it unit positive charge} is given by setting \(Q=1\) in \Eqref{EQ04}. Therefore the field due to the system of \(N\) charges is
\begin{eqnarray}\label{EQ05}
\vec{E}=\sum_k\left(\frac{q_k}{4\pi\epsilon_0}\right)
\frac{(\vec{r}-\vec{r_k})}{\mid\vec{r}-\vec{r}_k\mid^3}.
\end{eqnarray}

The above equation is just the statement of the superposition principle for the electric field.

The field of a point charge at large distance,\eqref{EQ03},  is given by
\begin{equation}
  \vec{E} \approx \frac{q_0}{4\pi\epsilon_0}\frac{(\vec{r}-\vec{r}_0)}{|\vec{r}-\vec r_0|^3}.
\end{equation} For a system of  localized charges, each term in the summation in \eqref{EQ05}, behave in the same manner and for large \(r\) we get  
\begin{eqnarray}\label{EQ06}
\vec{E}\approx \sum_k\left(\frac{q_k}{4\pi\epsilon_0}\right)\frac{(\vec{r}-\vec{r_k})}{\mid\vec{r}\mid^3}
= \frac{Q}{4\pi\epsilon_0}\frac{\vec r}{|\vec r|^3}.
\end{eqnarray}  where \(Q=\sum_k q_k\) is the total charge.
This is intuitively obvious that the system looks as if all charges are concentrated at the origin.
Thus the field at large distances approximates the field of a point charge \(Q\).

Rreferences

``What are Fields?''  Sec 1-5
R. P. Feynman, Robert B. Leighton and Mathew Sands{\it Lectures on Physics}, vol-II, B.I. Publications (1964)