Problem 17 - University Physics with Modern Physics

Question

The body contains many small currents caused by the motion of ions in the organs and cells. Measurements of the magnetic field around the chest due to currents in the heart give values of about 10 \(\mu\)G. Although the actual currents are rather complicated, we can gain a rough understanding of their magnitude if we model them as a long, straight wire. If the surface of the chest is 5.0 cm from this current, how large is the current in the heart?

Step-by-Step Solution

Answer

The current in the heart is approximately 500 µA.

Step 1: Identify the Known Values

We are given the magnetic field strength \( B = 10 \, \mu G = 10^{-5} \, G\) and it is converted to Tesla using \(1 \, G = 10^{-4} \, T\), so \( B = 10^{-9} \, T\). The distance \( r \) from the wire to the point of measurement is 5.0 cm, which is \(0.05 \, m\).

Step 2: Use Ampère's Law for a Long, Straight Wire

Ampère's Law for a magnetic field around a long, straight wire gives us the equation \( B = \frac{\mu_0 I}{2\pi r} \), where \( \mu_0 = 4\pi \times 10^{-7} \, T\cdot m/A\) is the permeability of free space.

Step 3: Solve for Current \( I \)

Rearrange the formula to solve for \( I \): \( I = \frac{2\pi r B}{\mu_0} \). Substitute the known values: \( I = \frac{2\pi (0.05) (10^{-9})}{4\pi \times 10^{-7}} \).

Step 4: Calculate the Current

Simplifying the calculation: \( I = \frac{2 \times 0.05 \times 10^{-9}}{4 \times 10^{-7}} \). The \( \pi \) terms cancel out, giving \( I = \frac{10^{-10}}{2 \times 10^{-7}} = 5 \times 10^{-4} \text{ A} \).

Step 5: Conclude the Result

The calculated current in the heart, modeled as a straight wire, is found to be \(5 \times 10^{-4} \text{ A}\) or \(500 \mu A\).

Key Concepts

Magnetic Field

Magnetic fields are an invisible force around a magnetic object or current-carrying wire. They are measured in units called Tesla (T) or Gauss (G), with 1 Gauss equal to 10\(^{-4}\) Tesla. These fields are generated by electric currents, which can be large or small, like those present in the human heart.
In the context of Ampère's Law, the magnetic field around a long, straight wire is directly proportional to the current through the wire. This means that as the current increases, so does the strength of the magnetic field around it. When measuring such fields, like around the chest due to the heart's currents, Metamaterials might help grasp the magnitude by modeling them as a straight wire.
This modeling simplifies complex biological systems to allow easier application of physical laws.

Current in the Heart

The heart functions with electrical impulses, causing small but significant currents. These currents are due to the motion of ions in the heart's cellular structures. While intricate in nature, these currents can be generalized to improve understanding.
This exercise models the heart's current as a long, straight wire, which simplifies calculations. The result found here indicates a current of about 500\(\mu \text{A}\) inside the heart. Such models are crucial for understanding electromagnetic fields in biological contexts, allowing simplifications that bring insight into the magnitude of these currents.

Permeability of Free Space

The permeability of free space, denoted as \(\mu_0\), is a fundamental constant in electromagnetism. It is approximately equal to \(4\pi \times 10^{-7} \, \text{T} \cdot \text{m/A}\). This constant is used in the formulation of Ampère's Law and other electromagnetic equations.
It signifies how much resistance the vacuum of space offers to the formation of a magnetic field. Thus, \(\mu_0\) connects magnetic field strength with physical parameters like current and distance. In the exercise, \(\mu_0\) helps compute the cardiac current from the measured magnetic field using the equation derived from Ampère's Law.

Straight Wire Model

The straight wire model simplifies complex current systems into a single, straight line. In physics, this model helps reduce elaborate systems to a manageable form for analytical purposes.
By treating the body's cardiac current as if it were flowing through a straight wire, one can use Ampère's Law for straightforward calculations. This model provides a practical approximation for the real-world scenarios where the exact path and distribution of current are highly sophisticated. For the given exercise, this model allows deriving meaningful insights about the current in the heart by comparing them to simpler systems.