Consider a blood-flow sensor placed around the artery in...

Consider a blood-flow sensor placed around the artery in Figure $\mathrm{P} 20.92 .$ Assume the artery has a radius of $3.0 \mathrm{mm}$ and the magnetic field is $B=0.050 \mathrm{T}$. a. If the blood speed is $25 \mathrm{cm} / \mathrm{s}$, what is the magnitude of the electric potential difference across the artery? b. Which side of the blood vessel in Eigure $\mathrm{P} 20.92$ is at a positive potential (relative to the other side)? c. How does the induced electric potential difference across the artery change if the artery is smaller, with $r=0.50 \mathrm{mm}$ d. Explain why this type of blood-flow meter is most useful for large arteries.

Consider a blood-flow sensor placed around the artery in Figure $\mathrm{P} 20.92 .$ Assume the artery has a radius of $3.0 \mathrm{mm}$ and the magnetic field is $B=0.050 \mathrm{T}$.
a. If the blood speed is $25 \mathrm{cm} / \mathrm{s}$, what is the magnitude of the electric potential difference across the artery?
b. Which side of the blood vessel in Eigure $\mathrm{P} 20.92$ is at a positive potential (relative to the other side)?
c. How does the induced electric potential difference across the artery change if the artery is smaller, with $r=0.50 \mathrm{mm}$
d. Explain why this type of blood-flow meter is most useful for large arteries.

Key Concepts

Electromagnetic Induction

This concept refers to the generation of an electromotive force (emf) or voltage across a conductor when it moves through a magnetic field. In the context of blood flow sensors, a conducting fluid like blood, when moving in a magnetic field, develops a potential difference across its cross-section. This is the fundamental principle that allows non?invasive measurement of blood flow using magnetic fields.

Lorentz Force

The Lorentz force is the force experienced by charged particles when they move in a magnetic field. In blood flow, the ions in blood are subjected to this force, which causes a separation of charges across the artery’s diameter. The resulting charge separation produces an electric potential difference proportional to the product of the blood velocity, the magnetic field strength, and the characteristic dimension of the vessel.

Magnetohydrodynamics (MHD)

Magnetohydrodynamics is the study of the dynamics of electrically conducting fluids in magnetic fields. It combines principles from both fluid dynamics and electromagnetism to explain how the moving, conductive blood interacts with an applied magnetic field. This interaction induces a voltage that can be measured and used to determine flow characteristics.

Dependence of Induced Voltage on Geometry

The induced voltage in a blood flow sensor is directly related to the dimensions of the artery, particularly its diameter. A larger vessel offers a greater distance over which the magnetic Lorentz force can separate charges, resulting in a larger potential difference. Conversely, in smaller arteries, the induced voltage is reduced due to the shorter span across which charge separation occurs.

Application of MHD in Blood Flow Measurement

MHD-based measurement techniques are used to determine blood flow by sensing the voltage induced by the moving, conductive blood in a magnetic field. This method is most effective in large arteries because the larger cross-sectional area yields a more significant voltage difference, making the measurement more reliable and easier to detect compared to smaller vessels.

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