Question

Define Gauge pressure and derive expression for it.

Answer 1

Hint: The degree by which the pressure measured in a fluid reaches the atmospheric pressure is known as gauge pressure. Hence, the SI unit is the same as that of pressure.

Complete answer:

The pressure measured in gauges is the pressure in relation to air pressure. For pressures above atmospheric pressure, gauge pressure is positive, and for pressures below atmospheric pressure, it is negative. Hence, Gauge pressure is the difference between the atmospheric pressure and the system pressure. Gauge pressure= $P - {P_a}$, where $P$ is the system pressure and ${P_a}$ is the atmospheric pressure.

Now, to derive the formula for gauge pressure let us consider a tank filled with water whose area is $A$ and height is $h$. We know that the pressure is calculated as Force per unit area.

$P = \dfrac{F}{A}$

Now, we know that the atmospheric pressure ${P_a}$  will depend on the weight of the atmosphere exerted on the surface. Basically, the atmospheric pressure is the weight of the gas molecules above a given area. Thus, in our case the force exerted on the bottom of the tank is due to the weight of the tank. Hence,

${P_a} = \dfrac{W}{A}$

$\Rightarrow {P_a} = \dfrac{{mg}}{A} $

We also know that the formula for density is 

$\rho  = \dfrac{m}{v} \\

\Rightarrow m = \rho v $

Substituting these values we get,

${P_a} = \dfrac{{\rho vg}}{A}$

$\Rightarrow {P_a} = \dfrac{{\rho {A}hg}}{{{A}}}$

$\therefore {P_a} = \rho hg$

Now, let us derive the gauge pressure formula. From the definition of the guage pressure formula we know that, the gauge pressure is given by:

Gauge pressure $= P - {P_a}$

Where,

$P$-The system pressure

${P_a}$- The atmospheric pressure

substituting the value of atmospheric pressure we get:

Gauge pressure $= P - {\rho}{g}{h}$

Hence, the formula to calculate Gauge pressure is $P - \rho hg$.

Note: There are various applications of gauge pressure. Its monitoring and control can be used in vacuum applications which precisely ventilate a vacuum chamber to atmospheric pressure and avoid particulate leakage when the chamber is opened to the atmosphere.