What is the gauge pressure of the water right at the point p, where the needle meets the wider chamber of the syringe? neglect the pressure difference caused by the radius change?

Missing details: figure of the problem is attached.

We can solve the exercise by using Poiseuille's law. It says that, for a fluid in laminar flow inside a closed pipe,

where:

is the pressure difference between the two ends

is viscosity of the fluid

L is the length of the pipe

is the volumetric flow rate, with  being the section of the tube and  the velocity of the fluid

r is the radius of the pipe.

We can apply this law to the needle, and then calculating the pressure difference between point P and the end of the needle. For our problem, we have:

is the dynamic water viscosity at 

L=4.0 cm=0.04 m

and r=1 mm=0.001 m

Using these data in the formula, we get:

However, this is the pressure difference between point P and the end of the needle. But the end of the needle is at atmosphere pressure, and therefore the gauge pressure (which has zero-reference against atmosphere pressure) at point P is exactly 3200 Pa.

magnified images possible images can be real = projected onto a screen