Problem 95 - Physics

Question

In a heart pacemaker, a pulse is delivered to the heart 81 times per minute. The capacitor that controls this pulsing rate discharges through a resistance of \(1.8 \times 10^{6} \Omega\). One pulse is delivered every time the fully charged capacitor loses \(63.2 \%\) of its original charge. What is the capacitance of the capacitor?

Step-by-Step Solution

Answer

The capacitance is approximately 0.4115 µF.

Step 1: Determine the pulse period

The pacemaker delivers 81 pulses per minute. First, find the time of each pulse: \[\text{Time per pulse} = \frac{60}{81} \text{ seconds/pulse} \approx 0.74074 \text{ s}\]

Step 2: Understand discharge behavior

A capacitor loses 63.2% of its charge at t = τ, when it discharges through a resistor. This corresponds to one time constant \(\tau\), where the charge decays according to the formula: \[\frac{Q}{Q_0} = e^{-t/\tau}\] For \(\frac{Q}{Q_0} = 0.368\), \(t = \tau\).

Step 3: Set up the RC time constant equation

The formula for the time constant \(\tau\) of an RC circuit is \[\tau = R \times C\] where \(R\) is the resistance and \(C\) is the capacitance. Here, \(\tau = 0.74074 \text{ s}\) and \(R = 1.8 \times 10^6 \underline{\phantom{xxx}} \Omega\).

Step 4: Solve for capacitance

Rearrange the equation \(\tau = R \times C\) to find \(C\): \[C = \frac{\tau}{R} = \frac{0.74074}{1.8 \times 10^6}\C \approx 4.115 \times 10^{-7} \text{ F}\]

Step 5: Convert to common units

Capacitance in microfarads (\(\mu F\)) is often more practical. Convert the result: \[C \approx 4.115 \times 10^{-7} \text{ F} \approx 0.4115 \underline{\phantom{xxx}} \mu F\]

Key Concepts

Understanding Capacitance Calculation

In simple terms, capacitance is a measure of a capacitor's ability to store charge. It's the ratio of the change in electric charge in response to the change in electric potential. In this pacemaker circuit scenario, the capacitor must discharge 63.2% of its charge to trigger a pulse. To find the capacitance, we need to understand the RC (resistor-capacitor) circuit dynamics.
In the exercise, a key formula is used which is the time constant equation:

• \[ \tau = R \times C \]
• Where \( \tau \) is the time constant, \( R \) is the resistance in ohms, and \( C \) is the capacitance in farads.

This equation tells us that the time constant \( \tau \) is directly proportional to both resistance \( R \) and capacitance \( C \). With the given resistance and a known time constant from the pacemaker timing, we can rearrange the formula to find capacitance \( C \). Understanding how to manipulate this basic formula is crucial in calculating capacitance in various circuit applications.

Explaining the Time Constant

The time constant, often represented by the Greek letter \( \tau \), is a critical concept in understanding circuit behavior. It represents the time it takes for the voltage across a capacitor to discharge to about 36.8% of its initial value, or conversely, to charge up to 63.2% of its full capacity.
In this problem, we saw how a pulse is triggered when the capacitor discharges to 36.8% of its initial charge. The time constant defines how quickly this discharge happens. For any RC circuit:

• Time constant \( \tau = R \times C \)
• It’s a unique parameter that determines the response speed of the circuit.
• For a pacemaker, the fast response is necessary to maintain a consistent pulse rhythm crucial for heart functioning.

Understanding the time constant helps electrical engineers design circuits that can control timing accurately, which is vital for devices like heart pacemakers.

Pacemaker Circuit Functions

A pacemaker is a medical device that uses electrical impulses to prompt the heart to beat at a normal rate. The pacemaker circuit includes a capacitor-resistor setup that controls the timing of these electrical pulses. Each pulse is crucial to maintaining regular heartbeat intervals.
The pacemaker delivers 81 pulses per minute, controlled by the RC circuit's time constant. By precisely engineering the RC time constant:

• Each capacitor discharge aligns with the required heartbeats per minute.
• Ensures reliability and precision necessary for heart health.

The capacitor charges to a predetermined value and upon discharge, sends an electrical signal to stimulate the heart, completing a vital cycle essential to life's rhythm. This understanding helps in designing reliable technology that can save lives by ensuring the heart beats in a stable, predictable pattern.

Charge Decay in Capacitors

Charge decay in capacitors is an important aspect of their function in circuits. The decay rate in a capacitor defines how long a capacitor will hold its charge before it drops to a specific level. The rate is exponential and represents the capacitor's discharge curve.
In mathematical terms, the charge \( Q \) at time \( t \) is given by the equation:

• \[ \frac{Q}{Q_0} = e^{-t/\tau} \]
• Where \( Q_0 \) is the initial charge.
• For a time equal to one time constant \( \tau \), the charge decreases to about 36.8%.

This is significant for the pacemaker because it needs the capacitor to systematically discharge to control pulse intervals accurately. By understanding this exponential decay, electrical engineers can predict and control how capacitors function in various circuit applications, crucial for precision timing devices like pacemakers.