Question

If the air density were uniform, then the height of the atmosphere above the sea level to produce a normal atmospheric pressure of $1.0 \times {10^5}\;{\text{Pa}}$ is ( density of air is $1.3\;{\text{kg/}}{{\text{m}}^{\text{3}}}$, $g=10\;{\text{m/}}{{\text{s}}^{\text{2}}}$ )
A) $0.77\;{\text{km}}$
B) $7.7\;{\text{km}}$
C) $77\;{\text{km}}$
D) $0.077\;{\text{km}}$

Answer 1

Hint: The pressure at any point inside the liquid directly proportional to the density of the liquid, acceleration due to gravity, and the depth of the point. Therefore as the depth increases the pressure will be increased.

Complete step by step answer:
The pressure can be defined as the force per unit area. The pressure is the force exerted on a body per unit area. This will be high when we go deep inside the sea. Hence the density of the liquid doesn’t vary. The pressure will be proportional to the height itself. And when we go above sea level the pressure will be very low.
At higher altitudes, the pressure is found to be very low. As the altitude increases the air that presses the body will be much smaller compared to the depths, hence the pressure of air will be less.
Given the density of air is $\rho = 1.3\;{\text{kg/}}{{\text{m}}^{\text{3}}}$ and atmospheric pressure is $P = 1.0 \times {10^5}\;{\text{Pa}}$.
The pressure at any point is given by the expression,
$
  P = \rho gh \\
  h = \dfrac{P}{{\rho g}} \\
$
Where, $\rho $ is the density, $g$ is the acceleration due to gravity and $h$ is the height.
Substituting the values in the above expression,
$
\Rightarrow h = \dfrac{{1.0 \times {{10}^5}\;{\text{Pa}}}}{{1.3\;{\text{kg/}}{{\text{m}}^{\text{3}}} \times 10\;{\text{m/}}{{\text{s}}^{\text{2}}}}} $
On simplification,
$\Rightarrow h = 7.69 \times {10^3}\;{\text{m}} $
$\Rightarrow h {\text{ = 7}}{\text{.7}}\;{\text{km}}$

Thus then the height of the atmosphere above the sea level is ${\text{7}}{\text{.7}}\;{\text{km}}$. The answer is option B.

Note:
We have to note that the depth is calculated from the top to the bottom of the sea level. Therefore the depth is smaller at high altitudes and larger at below the sea level. Thus the pressure increases with depth.