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Chapter 28: Problem 46

Connections to the body for electrocardiography (ECG) and electroencephalography (EEG) are normally made with metal electrodes and conductive gels to ensure good electrical contact. An alternative is the capacitively coupled noncontact electrode, which uses a conductor near but not contacting the skin, to form a capacitor. Clothing can serve as the capacitor's insulation, eliminating skin contact. A particular EEG instrument calls for capacitive electrodes with maximum reactance \(10 \mathrm{M} \Omega\) at a typical EEG beta wave frequency of 25 Hz. What's the minimum electrode capacitance?

Short Answer

Expert verified
To solve for the minimum electrode capacitance, calculate \(C = 1 / (2 \pi * 25 * 10*10^6) = 636.62 * 10^{-12} F = 636.62 pF\). So, the minimum electrode capacitance is approximately 636.62 pF.

Step by step solution

01

Understand the General Formula for Capacitive Reactance

Capacitive reactance, which is the resistance of a capacitor in an AC circuit, is given by the formula: \(X_C = 1 / (2 \pi f C)\), where \(X_C\) is the capacitive reactance, \(f\) is the frequency, and \(C\) is the capacitance. In this exercise, we're given \(X_C\) and \(f\), and we're being asked to solve for \(C\). Therefore, we'll need to rearrange the formula.
02

Rearrange the Formula

The goal is to isolate \(C\) in the formula. By doing some algebra, we can rearrange the formula as \(C = 1 / (2 \pi f X_C)\).
03

Substitute Given Values into the Formula

We're given that \(X_C = 10MΩ = 10x10^6 Ω\), and \(f = 25Hz\), so now we substitute these values into the formula: \(C = 1 / (2 \pi * 25Hz * 10x10^6 Ω)\).
04

Simplify the Formula

Upon substituting the given values into the formula, it's time to simplify. The result of the calculation gives us a numerical value for the capacitance.
05

Interpret the Result

Remember, this result represents the minimum electrode capacitance required for the EEG instrument to have a maximum reactance of 10MΩ at a frequency of 25Hz.

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