Measuring blood flow blood contains positive and negative...

Measuring blood flow. Blood contains positive and negative ions and therefore 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 Figure $21.62 . )$ (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 $\mathcal{E}=v B d .$ (b) If you expect that the blood will be flowing at 15 $\mathrm{cm} / \mathrm{s}$ for a vessel 5.0 $\mathrm{mm}$ in diameter, what strength of magnetic field will you need to produce a potential difference of 1.0 $\mathrm{mV} ?$ (c) Show that the volume rate of flow $(R)$ of theblood is equal to $R=\pi \mathcal{E} d / 4 B .$ (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 $\mathcal{E}$ must be made directly across the vessel.)

Transcript

00:01 Here we're going to look at something known as the hall effect, and it involves charges that are moving in a wire or a film, a very small structure, in the presence of a magnetic field.

00:20 The application is looking at blood flowing in a vessel, with the assumption that there are both positive and negative ions in that flow, of blood.

00:33 So i have drawn here a bloodstream, and we'll just assume for the sake of argument magnetic field pointing into the page perpendicular to the blood flow.

00:46 What we'll be using is the magnetic force on moving charges is equal to the charge and then v velocity crossed the magnetic field.

01:02 You may use your right hand and the right hand rule and figure out which direction the force would be on positive charges in that stream.

01:11 So i usually point my fingers on my right hand in the direction of the velocity and let them naturally curl, in this case, into the page.

01:22 And then my thumb is going to show the direction of the force on positive charges.

01:30 And and what i find out is that, of course, the positive charges are going to be pushed to the top of the tube.

01:38 They'll stay in the tube.

01:40 I'm just showing where the positive charges go.

01:44 And therefore, the negative ions will go down below.

01:50 So this charge separation will develop something called the haul voltage.

01:55 Haul voltage.

02:01 And that voltage will actually produce an electric force, of course.

02:08 So imagine that the top of the tube was a positive plate on a capacitor.

02:13 This is a very simplistic model, and the bottom is the negative plate.

02:19 So there is an electric field that points downwards.

02:24 And therefore, there is a force electrical force on the positive charges as well as the negative ones, q times e.

02:37 So there is a force trying to bring the charges back together again.

02:43 The haul voltage, meanwhile, is you can assume that there is a uniform field approximately, and delta v develops between the two ends and the strength.

02:59 Of the electric field is that haul voltage divided by the spacing.

03:05 So eventually those charges will collect and stop collecting charges, stop separating when there are enough charges on those two plates, or when the f -electric balances the force magnetic, just good old newton's law type stuff, balancing there.

03:37 So notice that the electric force on the positive charges is downwards and the magnetic force is upwards.

03:46 So that is what's happening.

03:48 There's an electric force that points down and a magnetic force that points up and those two cancel each other on a positive charge.

04:02 And of course they are opposite for the negative charge.

04:05 So everything works out really nice.

04:07 And so we can figure out the hall electric field.

04:18 And we'll call that the whole electric field and then the delta v haul.

04:29 So setting the electric force, qe equal to the magnetic force, notice that, yes, the charge cancels.

04:38 So it'll work for whatever ions are inside the blood.

04:44 And we can then put the relationship between potential and electric field into this relationship.

05:01 Now we want the dimensions of the blood vessel, which we're going to call d, the diameter.

05:08 So delta x is equal to the diameter of the tube, the blood vessel.

05:18 And so we have an expression for the whole voltage that builds up perpendicular to the magnetic field, dvb.

05:32 And so a question arises is, could we use that whole voltage as a way to measure blood flow in a patient maybe? i'm not sure how that would work, but the question is, how would this work in practice? so we're going to take a look at a couple practical cases...

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