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When we calculate electric field intensity for a point charge at any point inside electric field the field intensity is $E = F/q$ where $F$ is the force acting on charge $q$. In this case, the charge $q$ should be very small. The practical value of $q$ cannot be so small as needed. In defining electric field by measuring value $F/q$ is smaller than the actual value.
My question: How can the accurate value of electric field intensity be calculated?
I'm quite confused. Can someone point me out?
The electric field intensity is usually defined in introductory textbooks as the limit $$\lim_{q\rightarrow0}\frac{\mathbf F}{q},$$ i.e. the force on a test charge, per unit charge, when that test charge goes to zero. However, you are right in noting that the limit $q\rightarrow 0 $ is impossible to take because charge is quantized, and you can never have charges below one electron charge $q=e$.
Because of that, it is easier to define the electric field produced by a point charge $q_1$ at a relative separation $\mathbf r$ as the vector $$\mathbf E:=\frac{1}{4\pi\epsilon_0} \frac{q_1 \hat{\mathbf r}}{r^2},$$ and then stipulate as a physical law that the force such a field (or any electric field) exerts on a second charge $q_2$ is given by $\mathbf F=q_2\mathbf{E}$, with due stipulation that self-interactions are not allowed. This procedure is both mathematically consistent and concurs with physical, experimental observations.
