Question

Prove that energy per unit volume is pressure?

Answer 1

Answer:

To prove :

Energy per unit volume is pressure.

Calculation:

We will use dimensional analysis to prove that energy per unit volume is pressure :

LHS :

Energy per unit volume

$= \frac{energy}{volume}$

$= \frac{( {M}^{1} {L}^{2} {T}^{ - 2} )}{( {L}^{3}) }$

$= ({M}^{1} {L}^{ - 1} {T}^{ - 2} )$

RHS:

Pressure

$= \frac{force}{area}$

$= \frac{ ({M}^{1} {L}^{1} {T}^{ - 2} )}{ ({L}^{2}) }$

$= ({M}^{1} {L}^{ - 1} {T}^{ - 2} )$

So , LHS = RHS.

Hence , energy per unit volume is pressure.

Answer 2

$\huge{\underline{\underline{\red{\mathfrak{Answer :}}}}}$

As we know the following Dimension formula

$\Large{\sf{^{.} \: Energy \: = \: [ML^2T^{-2}]}}$

$\Large{\sf{^{.}\: Volume \: = \: [L^3] }}$

$\Large{\sf{^{.} \: Pressure \: = \: [ML^{-1}T^{-2}]}}$

$\rightarrow {\sf{\dfrac{[ML^2T^{-2}]}{[L^3]}}}$

$\rightarrow {\sf{[ML^{-1}T^{-2}}}$

So, ML^-1T^-2 is also equal to Pressure

We can say that pressure = Energy/Volume

$\large{\underline{\boxed{\sf{Pressure \: = \: \dfrac{Energy}{Volume} \: = \: \dfrac{Force}{Area}}}}}$