Estimate the pressure deep inside the sea at a depth `h` below the surface. Assume that the density fo water is `rho_(0)` at sea level and its bulk modulus is `B. P_(0)` is the atmosphere pressure at sea level `P` is the pressure at depth `'h'`

Options

A. `P = P_(0) - B In (1 - (rho_(0)gh)/(B))`
B. `P = P_(0) + B In (1 - (rho_(0)gh)/(B))`
C. `P = P_(0) - B In (1 + (rho_(0)gh)/(B))`
D. `P = P_(0) + B In (1 + (rho_(0)gh)/(B))`

Answer

A

Explanation

In a static fluid the pressure variation is given by
`(dP)/(dh) = -rho g` ...........(1)
The bulk modulus is defined as
`B = -(dP)/(dV//V)`......(2)
where `dV//V` is fractional change in volume of a element subjected to isotromic pressure incrases `dP`. We consider a sample of the fluid having mass `M`, its volume `V = M//rho` so that `dV = (M)/(rho^(2))d rho`
Hence `(dV)/(V)n = (d rho)/(rho)` ...........(3)
Combnineg equactions (2) and (3) we get
`(Bd rho)/(rho) = rho g dh`
or `int_(rho o)^(rho) (d rho)/(rho0^(2)) = int_(0)^(h) (gdh)/(B)` or `(1)/(rho_(0)) - (1)/(rho) = (gh)/(B)..............(4)`
Hence `int_(po)^(p) dP = itn_(rho o)^(rho) B (d rho)/(rho)` or `P - P_(0) = B` in `(rho)/(rho_(0)).........(5)`
On mutiplaing equaction (4) by `rho_(0)` we get
`1 - (rho_(0))/(rho) = (rho_(0) gh)/(B)`
So that In `(rho)/(rho_(0)) = -In (1 - (rho_(0)gh)/(B))`
Subsitituing this in equaction (5) we get
`P = P_(0) - B In (1 - (rho_(0) gh)/(B))`
This is the requird expression for `P`