Question

The dimension of $\dfrac{1}{2}\varepsilon_0 E^2$ is

Hint:

The expression $1/2 \epsilon_0 E^2$ is the formula for energy density (Energy/Volume).

Answer 1

Step 1: Identify the Physical Meaning. The expression $\frac{1}{2}\varepsilon_0 E^2$ is the formula for the energy density (u) of an electric field in a vacuum.

Step 2: Recall the formula for Energy Density. Energy density is defined as the amount of energy stored per unit volume: $u = \frac{\text{Energy}}{\text{Volume}}$.

Step 3: Substitute the Dimensions. The dimension of Energy (work) is $[ML^{2}T^{-2}]$. The dimension of Volume is $[L^{3}]$.

Step 4: Perform the Division. Dimension of $u = \frac{[ML^{2}T^{-2}]}{[L^{3}]} = [ML^{-1}T^{-2}]$.

Step 5: Verify. This matches the dimension of Pressure or Stress, which is also Force per unit Area ($MLT^{-2}/L^{2} = ML^{-1}T^{-2}$). Thus, the dimension of $\frac{1}{2}\varepsilon_0 E^2$ is $ML^{-1}T^{-2}$.