Question

To check the correctness of a physical equation by using dimensions, E=mc square​

Answer 1

Question :

To check the correctness of a physical equation by using dimensions, E = mc²

Solution :

In the Relationships E = mc²

• E is energy
• m is mass
• and c is speed of light

First, we will find Dimensional Formula of Energy :

Use formula for Potential Energy

$\longrightarrow \sf{E \: = \: mgh} \\ \\ \longrightarrow \sf{E \: = \: [M] [LT^{-2}] [L]} \\ \\ \longrightarrow \sf{E \: = \: [M] [L^2 T^{-2}]} \\ \\ \longrightarrow \sf{E \: = \: [ML^2 T^{-2}]} \\ \\ \underline {\underline{\sf{Dimensional \: Formula \: of \: Energy \: is \: [ML^2 T^{-2}]}}}$

$\rule{150}{1}$

Now, use the relationship E = mc²

$\longrightarrow \sf{[ML^2 T^{-2}] \: = \: [M] [LT^{-1}]^2} \\ \\ \longrightarrow \sf{[ML^2 T^{-2}] \: = \: [M] [L^2 T^{-2}]} \\ \\ \longrightarrow \sf{[ML^2 T^{-2}] \: = \: [ML^2 T^{-2}]} \\ \\ \longrightarrow \mathbb{L.H.S \: = \: R.H.S} \\ \\ \underline{\underline{\sf{The \: relation \: E \: = \: mc^2 \: is \: Dimensionally \: correct}}}$

Answer 2

Answer:

Given:

An equation has been provided such that :

E = mc²

To find:

If it's dimensionally correct or not

Calculation:

We shall try to see if the dimensions of LHS and RHS match or not .

LHS :

E = Energy

$E = force \times displacement$

$= > E = \{ML{T}^{ - 2} \times L \}$

$= > E = \{M {L}^{2} {T}^{ - 2} \}$

RHS:

m = mass and c = speed of light

$m \times {c}^{2} = \{ M \times {(L {T}^{ - 1} )}^{2} \}$

$= > m \times {c}^{2} = \{ M {L}^{2} {T}^{ - 2} \}$

LHS = RHS

So the equation is dimensionally correct.