Problem 22 - Physics

Question

In one measurement of the body's bioelectric impedance, values of \(Z=4.50 \times 10^{2} \Omega\) and \(\phi=-9.80^{\circ}\) are obtained for the total impedance and the phase angle, respectively. These values assume that the body's resistance \(R\) is in series with its capacitance \(C\) and that there is no inductance \(L .\) Determine the body's resistance and capacitive reactance.

Step-by-Step Solution

Answer

The body's resistance is approximately 444 Ω and its capacitive reactance is 76.52 Ω.

Step 1: Understand the Given Values

The total impedance \(Z\) of the body is given as \(4.50 \times 10^{2} \, \Omega\) and the phase angle \(\phi\) is \(-9.80^{\circ}\). We're asked to find the resistance \(R\) and the capacitive reactance \(X_C\) within the series RC circuit model. Knowing there is no inductance \(L\) simplifies the circuit to just \(R\) and \(C\) in series.

Step 2: Express Impedance in Rectangular Form

To determine the resistance and reactance, convert the impedance to rectangular form \(Z = R + jX_C\). The impedance in polar form is \(Z = |Z| e^{j\phi}\). Convert the phase angle to radians: \(-9.80^{\circ} = -0.17118\) radians.

Step 3: Calculate Resistance (R)

The resistance \(R\) is the real part of the impedance. This is calculated by \(R = |Z| \cos \phi\). Substitute \(|Z| = 4.50 \times 10^{2} \, \Omega\) and \(\phi = -0.17118\) radians. Thus, \(R = 4.50 \times 10^{2} \times \cos(-0.17118)\). Calculating gives \(R \approx 4.44 \times 10^{2} \, \Omega\).

Step 4: Calculate Capacitive Reactance (XC)

The capacitive reactance \(X_C\) is the imaginary part of the impedance. \(X_C = -|Z| \sin \phi\). Substitute the same values: \(X_C = -4.50 \times 10^{2} \times \sin(-0.17118)\). Calculating gives \(X_C \approx 76.52 \, \Omega\).

Step 5: Summary of Results

The calculations show that the resistance \(R\) is approximately \(444 \, \Omega\) and the capacitive reactance \(X_C\) is approximately \(76.52 \, \Omega\). These are the values consistent with the given impedance in the body.

Key Concepts

Resistance

Resistance is a crucial factor in understanding the flow of electricity through the body. It represents the opposition a material offers to the flow of electric current. In this exercise, resistance is one part of the body's bioelectric impedance, specifically the real part in the simple series RC circuit.When we talk about resistance in circuits like this, it's important to remember:

• Resistance (\(R\)) is always a positive value and is measured in ohms (\(\Omega\)).
• It reflects how much the material (such as body tissues) resists the passage of electrical current.
• In the impedance formula \(Z = R + jX_C\), \(R\) is the part without the 'imaginary' bit, also known as the real part of the total impedance.

It was identified by using the formula \(R = |Z| \cos \phi\), which underlays that resistance is linked with the total impedance(\(Z\)) and the phase angle(\(\phi\)).
This exercise highlights the practical aspect of resistance in measuring how one's body tissues impede electrical signals. Knowing \(R\) helps better interpret body compositions.

Capacitive Reactance

Capacitive reactance relates to the body's ability to store and release electrical energy, much like how a capacitor behaves in a circuit. This reactance can be thought of as the body's 'storage and switching' capability for current.In terms of calculation:

• Capacitive reactance (\(X_C\)) is also measured in ohms (\(\Omega\)).
• Unlike resistance, it considers the frequency of the electric signal and the capacitance of the material (or body tissue here).
• It usually shows up as an imaginary component in the impedance equation.

In this exercise, \(X_C\) was calculated using \(X_C = -|Z| \sin \phi\), showing how the phase angle (\(\phi\)) influences how electricity is stored within body tissue.
Through this calculation, it became clear that capacitive reactance helps describe how well electrical signals can pass through or be delayed by the body tissues.

Phase Angle

The phase angle in a bioelectric impedance context gives insight into the difference in phase between voltage and current when they pass through the body.A couple of key points about phase angle:

• Phase angle (\(\phi\)) is measured in degrees or radians. It's used to express the shift or delay in phase between current and voltage.
• In mathematical calculations, converting degrees to radians is necessary for trigonometric functions.
• It helps indicate proportions between resistance and reactance in a system.

In this exercise, knowing that \(\phi = -9.80^{\circ}\) gave a clear indication of how the capacitive properties of body tissue influenced the flow of electrical signals.
The phase angle acts as a fine-tuning measure in understanding body composition, often indicating health aspects, like cell membrane integrity.