Electric field

The interaction between charge \(Q\) and \(q\) is described by Coulomb's inverse-square law: \[ \vec{F}_{q} = \frac{1}{4 \pi \epsilon_{0}} \frac{Qq}{r^{2}} \hat{r}_{q} \] where \(\vec{F}_{q}\) is the force on charge \(q\) due to \(Q\) and \(\hat{r}\) is the unit vector along the position vector of \(q\) with respect to \(Q\). The electrostatic force on \(Q\) due to \(q\) is equal in magnitude but opposite in direction to \(\vec{F}_{q}\).

Everything we need to know to study interactions between static charges is contained in the Coulomb's law. However, in its present form the Coulomb's law does not lend itself to efficient use. For instance, say we want to study the effect of charge \(Q\) on \(q\). We will here employ Coulomb's to figure out the magnitude and direction of the force on \(q\) due to \(Q\). If we replace \(q\) with a different charge, say, \(q'\), we will have to evaluate the Coulomb's law all over again! We will need to repeat calculations every time we replace the \(Q\). This is not efficient. There is a way to avoid having to redo calculations. We can rearrange the equation for \(\vec{F}_{q}\) in the following manner

\[ \vec{F}_{q} = \left(\frac{1}{4 \pi \epsilon_{0}} \frac{Q}{r^{2}} \hat{r}_{q} \right) q \]

The term in the parentheses depends only on the charge \(Q\) and the position of \(q\). It is independent of the magnitude of \(q\). We may label this term as \(\vec{E}\). Because \(\vec{E}\) is independent of the charge \(q\), we can evaluate it at all points around \(Q\) without having to worry about the magnitude of \(q\). The force on \(q\) due to \(Q\) in terms of \(\vec{E}\) is \(\vec{F} = q\vec{E}\). If we replace the charge \(q\) at \(\vec{r}\) with a different charge, say, \(q'\), the force on \(q'\) due to \(Q\) can be obtained by simply multiplying \(\vec{E}(\vec{r})\) with the charge of \(q'\), i.e.

\[ \vec{F} = q' \vec{E}(\vec{r}) \]

If we replace \(q'\) with \(q''\), the force on \(q''\) is \[ \vec{F} = q'' \vec{E}(\vec{r}) \]

Thus, the onus of working through Coulomb's law every time we replace a charge has been greatly reduced, plus it makes the expression for force look nicer. The quantity \(\vec{E}(\vec{r})\) is referred to as the electric field due to charge Q at \(\vec{r}\). The concept of electric field is a very useful tool, especially when we want to analyze the effect of a system of fixed charge configuration on charges in its surrounding in which case we first evaluate electric field due to the system of fixed charges everywhere, and then use this field to study the effect of the system on the behavior of charges around it. This formulation of electrostatics in terms of electric field enables us to construct a rich framework of effective mathematics tools to solve complex problems. The concept of electric field lines illustrates this idea.

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