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# How Does a Barometer Work?

## Important Formula

## What do the formula symbols mean?

## Pressure

`$$ \mathit{\Pi} $$`Unit

`$$ \mathrm{Pa} = \frac{ \mathrm{N} }{ \mathrm{m}^2 } $$`

Air pressure (or the pressure of another fluid) at height \(h\) above the sea surface. According to this barometric formula, air pressure decreases exponentially with height.

## Pressure at the bottom

`$$ \mathit{\Pi}_0 $$`Unit

`$$ \mathrm{Pa} $$`

Air pressure at sea level \(h = 0\).

## Scale height

`$$ H $$`Unit

`$$ \mathrm{m} $$`

Scale height is a constant that depends, for example, on the temperature of the air and the gravitational acceleration. The scale height is approximately \(H=8\,\mathrm{km}\) on Earth.

## Height

`$$ h $$`Unit

`$$ \mathrm{m} $$`

Height measured from the sea surface.

A simple barometer works by using the **barometer formula**, which relates the height \(h\) above the ground to the air pressure \( \mathit{\Pi} \) prevailing at that height:

$$ \begin{align} \mathit{\Pi}(h) ~=~ \mathit{\Pi}_0 \, \mathrm{e}^{-h/H} \end{align} $$

Here \( H ~=~ 8005 \, \mathrm{m} \) and the mean air pressure at sea level is \(\mathit{\Pi}_0 = 101 \, \mathrm{kPa} = 1 \, \mathrm{bar} \). The formula states that the air pressure decreases *exponentially* with increasing height \( h \).

So an barometer measures the air pressure \( \mathit{\Pi} \) and calculates from it the height at which this air pressure prevails:

$$ \begin{align} h ~=~ 8005 \, \mathrm{m} \,\cdot\, \ln\left(\frac{ \mathit{\Pi} }{ 101 000 \, \mathrm{Pa} }\right) \end{align} $$