Question

The dimensional formula for electric field intensity is:
A. \[\left[ {ML{T^{ - 3}}{A^{ - 1}}} \right]\]
B. \[\left[ {ML{T^{ - 1}}{A^{ - 3}}} \right]\]
C. \[\left[ {ML{T^3}{A^{ - 1}}} \right]\]
D. \[\left[ {ML{T^{ - 3}}{A^1}} \right]\]

Answer 1

Hint: Use the formula for electric field intensity due to a charge. Convert the physical quantities in the formula for electric field intensity in the fundamental physical quantities. Substitute the dimensional formulae of all these fundamental physical quantities in the formula of electric field intensity and derive the dimensional formula for electric field intensity.

Formula used:
The electric field intensity \[E\] due a charge is given by
\[E = \dfrac{F}{q}\] ……. (1)
Here, \[F\] is the electric force acting on the charge and \[q\] is the charge.
The force \[F\] acting on a particle is
\[F = ma\] …… (2)
Here, \[m\] is the mass of the particle and \[a\] is acceleration of the particle.
The acceleration \[a\] of a particle is
\[a = \dfrac{v}{t}\] …… (3)
Here, \[v\] is the velocity of the particle and \[t\] is time.
The velocity \[v\] of a particle is
\[v = \dfrac{x}{t}\] …… (4)
Here, \[x\] is the displacement of the particle and \[t\] is time.
The charge \[q\] on a particle is given by
\[q = It\] …… (5)
Here, \[I\] is the current and \[t\] is time.

Complete step by step solution:
The electric field intensity due to a charged particle is the ratio of electric force acting on the particle to the charge on the particle.
\[E = \dfrac{F}{q}\]

To determine the dimensional formula for electric field intensity, all the physical quantities in the above formula must be converted in the form of seven fundamental physical quantities.

Substitute \[\dfrac{x}{t}\] for \[v\] in equation (3).
\[a = \dfrac{{\dfrac{x}{t}}}{t}\]
\[ \Rightarrow a = \dfrac{x}{{{t^2}}}\]

Substitute \[\dfrac{x}{{{t^2}}}\] for \[a\] in equation (2).
\[F = m\dfrac{x}{{{t^2}}}\]

Substitute \[m\dfrac{x}{{{t^2}}}\] for \[F\] and \[It\] for \[q\] in equation (1).
\[E = \dfrac{{m\dfrac{x}{{{t^2}}}}}{{It}}\]
\[ \Rightarrow E = \dfrac{{mx}}{{I{t^3}}}\] …… (6)
The dimensional formula for mass \[m\] is \[\left[ M \right]\].
The dimensional formula for displacement \[x\] is \[\left[ L \right]\].
The dimensional formula for electric current \[I\] is \[\left[ A \right]\].
The dimensional formula for time \[t\] is \[\left[ T \right]\].

Substitute for \[m\], for \[x\], for \[I\] and for \[t\] in equation (6).
\[ \Rightarrow E = \dfrac{{\left[ M \right]\left[ L \right]}}{{\left[ A \right]{{\left[ T \right]}^3}}}\]
\[ \Rightarrow E = \left[ M \right]\left[ L \right]\left[ {{A^{ - 1}}} \right]{\left[ T \right]^{ - 3}}\]
\[ \Rightarrow E = \left[ {ML{T^{ - 3}}{A^{ - 1}}} \right]\]

Therefore, the dimensional formula for electric field intensity is \[\left[ {ML{T^{ - 3}}{A^{ - 1}}} \right]\].

So, the correct answer is “Option A”.

Note:
We have derived the formula for electric field intensity due to a charge in the form of physical fundamental quantity in order to make the derivation of the required dimensional formula simpler. One can also directly substitute the dimensions of force and charge in the formula for electric field intensity.