Question

Why is the constant K in coulumbs law is eqaul to 1/4pie epsilon not......

and second que is that
why only K is used as constant of proportionality in coulumbs Law..??

Answer 1

1.This is because of gauss law which state that

close integral of Electrical field $E$ over area is equal to $\frac{q}{\epsilon}$

So,

Considering surface to be spherical

$\overset{s_1}{\int} E.ds=\frac{q}{\epsilon}=E\overset{s_1}{\int} ds$

$s_1=4\pi r^2$

$E.4\pi r^2=\frac{q}{\epsilon}$

$E=\frac{1}{4\pi \epsilon}\frac{q}{r^2}$

$F=q_{test}E=\frac{1}{4\pi \epsilon}\frac{q_{test}q}{r^2}$

$k=\frac{1}{4\pi \epsilon}$

So, value of $k$ $=\frac{1}{4\pi \epsilon}$

2.

Let 2 charges $q_1,q_2$ are seperate by $r$ distance

Accrding to coulomb law

$F$ is directly proportional to $q_1q_2$

$F$ is directly proportional to $\frac{1}{r^2}$

combining both

$F$ is directly proportional to $\frac{q_1q_2}{r^2}$

removing proportionality sign by putting a constant $k$

$F=\frac{kq_1q_2}{r^2}$

So. only $k$ is used as constant of proportionality in coulomb law