Question

The dimension of $\dfrac{1}{2}\epsilon_oE^2$ .where E is the electric field and $\epsilon _o$ is the permittivity of free space.
$\begin{align}
  & A.[M{{L}^{2}}{{T}^{-1}}] \\
 & B.[M{{L}^{-1}}{{T}^{-2}}] \\
 & C.[M{{L}^{2}}{{T}^{-2}}] \\
 & D.[ML{{T}^{-1}}] \\
\end{align}$

Answer 1

Hint: Checking if the given expression is a formula of any quantity, we come to know that the given is the expression for energy per unit volume. Energy per unit volume can be written in terms of mass length and time.

Step by step solution:
We need to identify if the given expression is a formula for any quantity. If yes, then we can find the dimension of the quantity easily by writing the formula in known terms as below :

The given expression is a formula for energy per unit volume. We can write energy and volume in terms of length mass and time (fundamental quantities) so we get

Energy can be written as work done which is equal to force times displacement. So we can

Write energy = work done = force $\times$ displacement
Force = mass $\times$ acceleration

So we can write units for energy as

Energy = work done = force$\times$ displacement = mass$\times$ acceleration$\times$ displacement = $kg\times\dfrac{m}{s^2}\times m$

Similarly volume can be written as Volume = length$\times$ breadth$\times$ height = $m^3$

The units of energy per unit volume can be now written as

$\dfrac{ kg\times\dfrac{m}{s^2}\times m}{m^3} = [ML^{-1}T^{-2}]$ . where L is the length , M is the mass and T is the time.

Note: This type of question can be solved by identifying the formula having the given expression and then expressing the formula in terms of fundamental quantities length, mass and time.