An electromagnetic flowmeter is used to measure blood flow...

An electromagnetic flowmeter is used to measure blood flow rates during surgery. Blood containing ions (primarily $\mathrm{Na}^{+}$ ) flows through an artery with a diameter of $0.50 \mathrm{~cm}$. The artery is in a magnetic field of $0.35 \mathrm{~T}$ and develops a Hall voltage of $0.60 \mathrm{mV}$ across its diameter. (a) What is the blood speed (in $\mathrm{m} / \mathrm{s}$ )? (b) What is the flow rate $\left(\mathrm{in} \mathrm{m}^{3} / \mathrm{s}\right) ?(\mathrm{c})$ If the magnetic field points west and the blood flow is north, is the top or bottom of the artery at the higher potential?

Key Concepts

Electromagnetic Flowmeter

An electromagnetic flowmeter is a device that uses the principles of electromagnetic induction to measure the flow rate of a conductive fluid. When a conductive fluid passes through a magnetic field, an electric potential is induced perpendicular to both the direction of flow and the magnetic field. This induced voltage is proportional to the speed of the fluid, allowing for accurate flow measurements without any moving parts.

Magnetic Induction

Magnetic induction refers to the process where a changing magnetic environment creates an electrical potential or electromotive force (emf). This is the principle behind electromagnetic flowmeters, where the conductive fluid moving through a static magnetic field induces a voltage proportional to its velocity through Faraday’s law of induction.

Hall Effect

The Hall effect is observed when a magnetic field is applied perpendicular to the flow of electric charge in a conductor, creating a measurable transverse voltage. In the context of flow measurement, this effect explains the development of a potential difference (Hall voltage) across the fluid channel, which is used to determine the fluid’s speed.

Fluid Velocity Calculation

The fluid velocity in electromagnetic flowmeters is determined using the relation v = E/B, where v is the fluid velocity, E is the induced electric field (directly related to the measured Hall voltage), and B is the magnetic field strength. This relationship is fundamental in converting the electrical measurements into a meaningful physical flow rate.

Flow Rate Determination

Flow rate quantifies the volume of fluid passing through a cross-sectional area per unit time and is calculated as the product of the fluid velocity and the cross-sectional area of the conduit. This calculation is essential in many applications, including medical settings, where accurate measurement of blood flow is critical.

Direction of Induced Potential

The determination of which part of the fluid's cross-section is at a higher potential involves understanding the right-hand rule for cross products, where the direction of the induced emf (and thus the potential distribution) depends on both the direction of the magnetic field and the direction of the fluid flow. This concept is crucial for interpreting measurements in systems where directional flow information is important.

Transcript

00:02 Hi, in the given problem, suppose here this is the artery.

00:09 In this artery, the blood is flowing towards right or north towards north.

00:19 So this is south, so upward, it will be west and here it will be east.

00:27 In the direction of flow of the blood, the sodium ions, positive ions are also moving.

00:34 Towards north and a magnetic field has been applied due west like this.

00:46 This is the velocity of the blood flow.

00:51 Now the diameter of the artery will serve as the length of the conductor which is moving in the magnetic field.

01:05 So, diameters, of the artery serves as length of wire, length of a wire or of a conductor.

01:28 Hence this length or the diameter of the artery is 0 .50 centimeter.

01:37 The magnetic field is given as 0 .35 tesla.

01:43 And the whole voltage developed across this diameter which is actually the motional emf induced and that has been given as v is equal to 0 .60 millivolt and we know charge over the sodium ion is 1 .6 into 10 times per minus 19 cullum.

02:28 Now, in the first part of the problem, we have to find the velocity of the blood flow.

02:36 So using the expression for motionally mf induced across the ends of a conductor, when it moves such that it intercepts the magnetic field lines perpendicularly.

02:48 And here also this is happening...

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