Question

Derive a dimensional formula for

COULOMB'S CONSTANT.

Answer 1

Determination of the dimensional formula for Coulomb's constant:

According to Coulomb's law; the electrostatic force between two point charges placed at a distance is given by;

$F=k_e\frac{q_1{q}_2}{r^2}$

Where $k_e$ is Coulomb's constant.

Now,

$F=k_e\frac{q_1{q}_2}{r^2}$

$k_e=\frac{F{r}^2}{q_1{q}_2}$

The dimension of

$k_e$ is given as;

$dimensionof{k}_e=\frac{dimensionofF\times dimensionof{r}^2}{dimensionof{q}_1\times dimensionof{q}_2}$

$dimensionof{k}_e=\frac{\left[ML{T}^{-2}\right]\left[L^2\right]}{\left[IT\right]\left[IT\right]}Or\frac{\left[ML{T}^{-2}\right]\left[L^2\right]}{\left[AT\right]\left[AT\right]}$

$dimensionof{k}_e=\frac{\left[M{L}^3{T}^{-2}\right]}{\left[I^2{T}^2\right]}Or,\frac{\left[M{L}^3{T}^{-2}\right]}{\left[A^2{T}^2\right]}$

$dimensionof{k}_e=\left[M{L}^3{I}^{-2}{T}^{-2-2}\right]Or,\left[M{L}^3{A}^{-2}{T}^{-2-2}\right]$

$dimensionof{k}_e=\left[M{L}^3{I}^{-2}{T}^{-4}\right]Or,\left[M{L}^3{A}^{-2}{T}^{-4}\right]$

Thus,

The dimension of Coulomb's constant is $\left[M{L}^3{I}^{-2}{T}^{-4}\right]$ or $\left[M{L}^3{A}^{-2}{T}^{-4}\right]$.