Question

The dimension of $\frac{1}{2} \epsilon_0 E^2$ is $M^a L^b T^c$ then value of $a - 2b + c = ?$

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

The Correct Option is A

Step 1: Identify the given expression.
The expression is $\frac{1}{2} \epsilon_0 E^2$.

Step 2: Understand the physical meaning of the expression.
In electromagnetism, the energy stored per unit volume in an electric field $E$ is known as the energy density. The formula for energy density $u_E$ is given by:
$ u_E = \frac{1}{2} \epsilon_0 E^2 $
where $\epsilon_0$ is the permittivity of free space and $E$ is the electric field strength.

Step 3: Determine the dimensions of energy density.
Energy density is defined as energy per unit volume.
$ [\text{Energy density}] = \frac{[\text{Energy}]}{[\text{Volume}]} $
The dimensions of energy are $[ML^2T^{-2}]$.
The dimensions of volume are $[L^3]$.

Step 4: Calculate the final dimensional formula.
$ [\frac{1}{2} \epsilon_0 E^2] = \frac{[ML^2T^{-2}]}{[L^3]} = [ML^{-1}T^{-2}] $

Step 5: Compare with the given form $M^a L^b T^c$.
By comparing $[M^1 L^{-1} T^{-2}]$ with $[M^a L^b T^c]$, we get:
$a = 1$
$b = -1$
$c = -2$

Step 6: Calculate the required value $a - 2b + c$.
Substituting the values of $a$, $b$, and $c$:
$ a - 2b + c = 1 - 2(-1) + (-2) $
$ a - 2b + c = 1 + 2 - 2 = 1 $

Final Answer: The value is 1. Correct option is (1).