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Chapter 27: Problem 75

One way to measure blood flow when blood vessels are exposed during surgery is to use an electromagnetic flow meter. This device surrounds the blood vessel with an electromagnet, creating a magnetic field perpendicular to the blood flow. since blood is a modest conductor, a motional emf develops across the blood vessel. Given vessel diameter \(d\), magnetic field \(B\), and voltage \(V\) measured across the vessel, show that the volume blood flow is given by \(\pi d^{2} V / 4 B d.\)

Short Answer

Expert verified
The volume blood flow is given by \( Q = \pi d^2 V / (4 \cdot B \cdot d) \).

Step by step solution

01

Understand Motional EMF

Motional emf is generated when a conductor moves in a magnetic field and it is given by \( E = B \cdot l \cdot v \), where \( B \) is the magnetic field strength, \( l \) is the length of the conductor, and \( v \) is the velocity of the conductor.
02

Apply Motional EMF to the Blood Vessel

In our case, the blood moving in the blood vessel acts as the 'moving conductor'. We are given that the voltage \( V \) measured across the vessel is the induced emf, thus we write the equation for motional emf as: \( V = B \cdot d \cdot v \) where \( v \) is the velocity of the blood flow.
03

Solve for Blood Velocity

From the motional emf expression established in step 2, we can solve for the blood velocity \( v \): \( v = V / (B \cdot d) \)
04

Establish Flow Rate Concept

In fluid dynamics, the volume flow rate (Q) of a fluid is given by: \( Q = A \cdot v \), where \( A \) is the cross sectional area of the path and \( v \) is the velocity of the flow. Since the blood vessel can be assumed to be cylindrical, its cross-sectional area \( A \) is: \( A = \pi (d/2)^2 \)
05

Derive the Equation for Volume Blood Flow

Substituting the expressions for \( v \) and \( A \) into the volume flow rate equation: \( Q = \pi (d/2)^2 \cdot V / (B \cdot d) \), and simplifying, gives us the volume blood flow equation: \( Q = \pi d^2 V / (4 \cdot B \cdot d) \)

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Most popular questions from this chapter

The magnetic field inside a 20 -cm-diameter solenoid is increasing at 2.4 T/s. How many turns should a coil wrapped around the outside of the solenoid have so that the emf induced in the coil is \(15 \mathrm{V} ?\)

A generator consists of a rectangular coil \(75 \mathrm{cm}\) by \(1.3 \mathrm{m},\) spinning in a 0.14 -T magnetic field. If it's to produce a \(60-\mathrm{Hz}\) alternating emf with peak value \(6.7 \mathrm{kV},\) how many turns must it have?

A 2.0 -A current is flowing in a \(20-\mathrm{H}\) inductor. A switch opens, interrupting the current in 1.0 ms. Find the induced emf in the inductor.

Can an induced electric field exist in the absence of a conductor?

You're an electrical engineer designing an alternator (the generator that charges a car's battery). Mechanical engineers specify a 10-cm-diameter rotating coil, and you determine that you can fit 250 turns in this coil. To charge a 12 -V battery, you need a peak output of \(14 \mathrm{V}\) when the alternator is rotating at \(1200 \mathrm{rpm}\). What do you specify for the alternator's magnetic field?
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