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Measuring Blood Flow. Blood contains positive and negative ions and thus is a conductor. A blood vessel, therefore, can be viewed as an electrical wire. We can even picture the flowing blood as a series of parallel conducting slabs whose thickness is the diameter d of the vessel moving with speed v. (See Fig. E29.34.) (a) If the blood vessel is placed in a magnetic field B perpendicular to the vessel, as in the figure, show that the motional potential difference induced across it is E = vBd. (b) If you expect that the blood will be flowing at 15 cm/s for a vessel 5.0 mm in diameter, what strength of magnetic field will you need to produce a potential difference of 1.0 mV? (c) Show that the volume rate of flow (R) of the blood is equal to R = πEd/4B. (Note: Although the method developed here is useful in measuring the rate of blood flow in a vessel, it is limited to use in surgery because measurement of the potential E must be made directly across the vessel.)

Short Answer

Expert verified

(a) If the blood vessel is placed in a magnetic field B which is perpendicular to the vessel, hence it is shown that the motional potential difference induced across idε=Bvd

(b) If the blood flows at 15cm/ s for the vessel diameter, the magnetic field needed to produce a potential difference of is

(c) The rate of blood flow is proven to beR=πεd4B

Step by step solution

01

Proving ε=Bvd

The blood slab has a flowing blood which has a width of d and is perpendicular to the field with a speed of v, which is equivalent to a rod of length d, and is moving in a magnetic field B, hence the induced emf is:ε=Bvd

02

Calculating the magnetic field

The blood is flowing at a speed of v = 15cm /s, vessel with diameter d = 5.0mm, and the potential difference ofε=1.0mV . Therefore, the magnetic field is given by:

B=εvd=1.0×10-3v0.15m/s5.0×10-3m=1.3T

03

Rate of flow of blood

The cross-sectional area of blood vessel is:

A=πd24

Now, the volume of blood flowing past the cross-section of the vessel equals to the distance that is travelled in time t is (let), therefore:

v=πd2x4

Since the blood is moving with a velocity of v, then x = vt therefore the equation becomes:

V=(πd2vt)4

Therefore, the volume rate is:

R=Vt=πd2v4=πd2v4εBd=πεd4B=πεd4B

Hence, proved that blood flow isπεd4B

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