2) The following describes a fluid storage device having the capability of measu

Question

2) The following describes a fluid storage device having the capability of measuring the height of a column of fluid stored within the device (see figure). The device includes an inner cylindrical solid conductor having radius A and negligible resistivity. The device also includes an outer cylindrical conducting shell (negligible resistivity) with inner radius B>A such that the wire and the shell are vertically concentric. An insulating end-cap (infinite resistivity) seals the bottom end whileH the top end-cap is open. The height of the shell/wire is H and the fluid can be poured into the open end-cap and stored within annular space between the conductors. The height of the fluid isy and the device is full when y . An ideal source of EMF E) is connected between the center conductor (positive terminal) and the shell (negative terminal). The fluid obeys Ohm's Law and has finite resistivity, p. Current I (measured by an ammeter) flows radially through the fluid from the wire to the conducting shell. The given quantities are A, B, H, and p a) Determine an expression for the resistance between the wire and the conducting shell. The expression should include the height y b) Show that he current flowing through the fluid is proportional to the height of the fluid c) Determine an expression for the constant of proportionality (relating to part b). d) Determine an expression for the potential difference between the outer conducting shell and a point within the Buid (defined as point P), such that P is a distance r from the center of the inner conducting solid and such that A

Explanation / Answer

given, inner radius = A

outer radius = B

height of liquid inside the apparatus = y

rstivity of liquid = rho

total height = H

a. for this apparatus

resistance of the appratus = R

consider some radius r, such that A < r < B

then consider thickness dr of the liquid

resistance of this small amount, dR = rho*dr/2*pi*r*y

total resistance of the liquid in the apparatus = integrate dR from r = A to r = B

R = rho*ln(B/A)/2*pi*y

b. now, EMF = E

using ohms law

E = iR

i = E/R = 2*pi*y*E/rho*ln(B/A)

hence i is p[ropoortional to y]

c. constant of proportionality in previous question is 2*pi*E/rho*ln(B/A)

d. for A < r < B

reisstance of the fluid taken from r = r to r = B is

R' = rho*ln(B/r)/2*pi*y

hence potential difference

V = i*R' = 2*pi*y*E*rho*ln(B/r)/2*pi*y*rho*ln(B/A)

V = E*rho*ln(B/r)/ln(B/A)

e. current density at this point P is i/2*pi*ry

J = 2*pi*y*E/rho*ln(B/A)*2*pi*r*y = E/rho*ln(B/A)r

f. electric field at P = -dV/dr = -d(E*rho*ln(B/r)/ln(B/A))/dr = E*rho/rln(B/r)

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