Question

What are the dimensions of the constant k in Coulomb’s law of electrostatics?

a. ML^2T^−4T^−2A^−1

ML^2T^3A^−2

M^−2L^3T^2A^−1

ML^3T^−4A^−2

Answer 1

What are the dimensions of the constant $k$ in Coulomb's law of electrostatics?

a. $ML^{2}T^{-4}T^{-2}A^{-1}$

b. $ML^{2}T^{3}A^{-2}$

c. $M^{-2}L^{3}T^{2}A^{-1}$

d. $ML^{3}T^{-4}A^{-2}$

Coulomb's law of electrostatics:

$$F = \frac {k (q _ {1} q _ {2})}{r ^ {2}}$$

where $F$ – force, $q$ – charge, $r$ – distance.

Therefore $k$ equals:

$$k = \frac {F r ^ {2}}{q _ {1} q _ {2}}$$

dimension of force is N, dimension of distance is meter, dimension of charge is C.

Newton's second law of motion: $F = m * a$, therefore dimension of N:

$$[ N ] = \left[ k g * \frac {m}{s ^ {2}} \right]$$

And, by definition: $I = \frac{dq}{dt}$, therefore dimension of C: $C = A * s$

Finally, for $k$ we have:

$$[ k ] = \left[ k g * \frac {m}{s ^ {2}} * \frac {m ^ {2}}{(A * s) ^ {2}} \right] = \left[ \frac {k g m ^ {3}}{s ^ {4} A ^ {2}} \right]$$

So, correct answer is d. $ML^{3}T^{-4}A^{-2}s$

Answer: d.