Problem 26 - College Physics

Question

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.)

Step-by-Step Solution

Answer

(a) \(\mathcal{E} = vBd\); (b) \(B = 0.0133 \, \text{T}\); (c) \(R = \pi \mathcal{E} d / 4B\).

Step 1: Understand the Given Variables

We have a blood vessel with diameter \(d\), flowing at a speed \(v\), placed in a perpendicular magnetic field \(B\). The task is to derive the formula for the motional potential difference and solve subsequent problems related to it.

Step 2: Derivation of Motional Potential Difference

The motional potential difference \(\mathcal{E}\) is given by the formula \(\mathcal{E} = vBd\). This formula arises because a conducting slab moving in a magnetic field experiences electromagnetic induction, creating a potential difference proportional to the speed, magnetic field strength, and diameter of the vessel.

Step 3: Calculate the Required Magnetic Field

We need to find \(B\) so that \(\mathcal{E} = 1.0 \, \text{mV}\) with \(v = 15 \, \text{cm/s}\) and \(d = 5.0 \, \text{mm}\). Using \(\mathcal{E} = vBd\), we substitute the values: \(1.0 \, \text{mV} = 15 \, \text{cm/s} \times B \times 0.5 \, \text{cm}\). Solving for \(B\):\[ B = \frac{1.0 \, \text{mV}}{15 \, \text{cm/s} \times 0.5 \, \text{cm}} = \frac{0.001 \, \text{V}}{0.075 \, \text{m/s}} = 0.0133 \, \text{T}\]

Step 4: Derivation of Volume Rate of Flow

The volume rate of flow \(R\) is derived from the equation \(R = \pi \mathcal{E} d / 4B\). This follows from the assumption that the volume rate is related to the cross-sectional area of the vessel and the flow speed, with the potential difference \(\mathcal{E}\) providing an indirect measurement of flow speed.

Key Concepts

Blood Flow Measurement

The measurement of blood flow is a fascinating application of electromagnetic induction. Blood can conduct electricity due to the presence of charged ions. This conductivity allows blood vessels to be treated like electrical wires. When we consider blood flowing through a vessel, it behaves like slabs moving with a certain velocity. By placing this vessel in a magnetic field that is perpendicular to the flow, we can apply principles of electromagnetism to measure flow characteristics.
In practical terms, when blood flows through a magnetic field, a potential difference is induced across the vessel. This occurs because of the movement of charged particles in the blood within the magnetic field. This induced potential difference can be used to measure how fast the blood is flowing — an essential parameter in medical diagnostics.
However, this method's practical use is primarily restricted to scenarios where direct access to the vessel is possible, such as during surgery, due to the need to measure the potential directly across the vessel.

Motional Potential Difference

The concept of motional potential difference is at the heart of electromagnetic induction processes like blood flow measurement. When a conductor moves through a magnetic field, an electromotive force (emf) is induced. This can be understood via Fleming's right-hand rule, which describes the direction of induced current when a conductor moves in a magnetic field.
For a blood vessel, the magnitude of this induced potential difference is given by the formula \( \mathcal{E} = vBd \) where:

• \( \mathcal{E} \) is the motional emf or potential difference,
• \( v \) is the velocity of blood flow,
• \( B \) is the magnetic field strength,
• \( d \) is the diameter of the vessel.

This formula shows that the induced potential is directly proportional to the flow velocity, the magnetic field's strength, and the vessel's diameter. Therefore, knowing any three of these variables allows us to compute the fourth, providing crucial information about blood flow very effectively.

Magnetic Field in Circuits

In the context of blood flow measurement, the magnetic field acts as the key agent in the induction of motional emf. A uniform magnetic field applied perpendicular to the movement of charges in the blood creates conditions necessary for maximizing emf.
The strength of the magnetic field is crucial. Too weak a field might not induce a measurable potential difference, while too strong a magnetic field is impractical for medical settings. The strength required can be calculated based on the desired potential difference for instrument readings.
For instance, if we need a potential difference of 1.0 mV with a given blood speed and vessel diameter, the magnetic field strength \( B \) can be calculated using the formula \[ B = \frac{\mathcal{E}}{vd} \].By substituting the values, one can determine the appropriate field strength needed to achieve accurate measurements without compromising patient safety. This precise application of magnetic fields showcases the intersection of physics and medicine, offering non-invasive techniques for monitoring crucial health parameters.