Question

A capacitor is a device that stores electrical charge. The charge on a capacitor accumulates according to the function \(Q(t)=a\left(1-e^{-t / c}\right),\) where \(t\) is measured in seconds, and \(a\) and \(c>0\) are physical constants. The steady-state charge is the value that \(Q(t)\) approaches as \(t\) becomes large. a. Graph the charge function for \(t \geq 0\) using \(a=1\) and \(c=10\) Find a graphing window that shows the full range of the function. b. Vary the value of \(a\) while holding \(c\) fixed. Describe the effect on the curve. How does the steady-state charge vary with \(a ?\) c. Vary the value of \(c\) while holding \(a\) fixed. Describe the effect on the curve. How does the steady-state charge vary with \(c ?\) d. Find a formula that gives the steady-state charge in terms of \(a\) and \(c\)

Short answer

Short Answer: When varying the value of \(a\), it affects the maximum steady-state charge that the function approaches. Increasing \(a\) increases the maximum charge, while decreasing \(a\) decreases it. When varying the value of \(c\), it affects the time it takes for the charge to reach the steady-state, with larger values of \(c\) making the process slower. The formula for the steady-state charge in terms of \(a\) and \(c\) is simply \(Q_{ss}=a\), which means that it only depends on the value of \(a\) and not \(c\).

Step-by-step solution

Set values and graph the function

With the given values \(a=1\) and \(c=10\), the function becomes \(Q(t)=1\left(1-e^{-t/10}\right)\). Create a graph of this function for \(t \geq 0\) with an appropriate window to display the steady-state charge.

Suitable graphing window

A suitable graphing window should show the behavior of the function approaching the steady-state charge when \(t\) becomes large. A window with \(0 \leq t \leq 50\) and \(0 \leq Q(t) \leq 1.2\) should work well in this case. #b) Varying the value of a on the graph and steady-state charge#

Vary the value of a and analyze the graph

Keeping the value of \(c\) constant, change the value of a and create new graphs showing the altered function. Compare these graphs to understand how varying \(a\) affects the shape of the curve.

Determining the effect of a on the steady-state charge

Analyze the steady-state charge behavior in the new graphs and infer the relationship between \(a\) and the steady-state charge. #c) Varying the value of c on the graph and steady-state charge#

Vary the value of c and analyze the graph

Keeping the value of \(a\) constant, change the value of \(c\) and create new graphs showing the altered function. Compare these graphs to understand how varying \(c\) affects the shape of the curve.

Determining the effect of c on the steady-state charge

Analyze the steady-state charge behavior in the new graphs and infer the relationship between \(c\) and the steady-state charge. #d) Steady-state charge formula in terms of a and c#

Find limit of Q(t) as t approaches infinity

To find the formula for the steady-state charge, find the limit of the function \(Q(t)=a\left(1-e^{-t/c}\right)\) as \(t\) approaches infinity. $$\lim_{t\to\infty} Q(t)=\lim_{t\to\infty} a\left(1-e^{-t/c}\right)$$

Steady-state charge formula

After calculating the limit, find the formula representing the steady-state charge in terms of \(a\) and \(c\).

Key concepts

Steady-State Charge

When a capacitor is charging, it doesn't just immediately reach its maximum capacity. Instead, it gradually builds up the charge until it reaches what is known as the steady-state charge. The steady-state charge is essentially the maximum capacity of the capacitor once it is fully charged. In the function \(Q(t) = a(1 - e^{-t/c})\), the charge approaches \(a\) as \(t\) (time) becomes very large. This means that as time progresses endlessly, \(Q(t)\) gets closer and closer to \(a\), since \(e^{-t/c}\) approaches zero. Thus, the steady-state charge is dictated by the constant \(a\).

The steady-state charge is calculated using a limit: \(\lim_{t\to\infty} Q(t) = a\). By applying limits to the function, the exponential part \(e^{-t/c}\) trends toward zero, confirming that the charge \(Q\) will reach the value \(a\). This relationship outlines the importance of the variable \(a\) and indicates why it serves as the determining factor for the steady-state charge.

Graphing Functions

Graphing the function \(Q(t) = a(1 - e^{-t/c})\) is essential for visualizing how the charge on a capacitor accumulates over time. Initially, the function starts at \(Q(0) = 0\) when no charge is present. As time \(t\) progresses, \(Q(t)\) rises and gets closer to the steady-state charge \(a\).
To capture this behavior graphically, ensure your graphing window shows the range of \(t\) values up to where the curve flattens out. Typically, a window with \(0 \leq t \leq 50\) and \(0 \leq Q(t) \leq a + 0.2\) would be appropriate. This allows you to observe the approach to the steady-state on the curve.

• When varying \(a\), the height of the steady-state charge changes. Larger \(a\) increases the final charge on the capacitor.
• When varying \(c\), the rate at which the charge approaches the steady-state changes. A larger \(c\) makes the approach slower, flattening the curve earlier on the timeline.

Graphing functions like these illustrates these relationships effectively, providing a clear picture of how different parameters affect capacitor charging dynamics.

Exponential Decay

Understanding exponential decay is crucial in analyzing the capacitor's charge function \(Q(t) = a(1 - e^{-t/c})\). Exponential decay describes how quantities decrease rapidly at first, then slow down as they near a stable level. In our context, it applies to \(e^{-t/c}\) which decreases as time \(t\) increases.

This decay is the reason that \(Q(t)\) eventually approaches \(a\). Initially, when the capacitor starts charging, \(e^{-t/c}\) is significant, causing rapid change in \(Q(t)\). However, as time elapses, \(e^{-t/c}\) diminishes, slowing the change in \(Q(t)\) until it virtually stops when reaching \(a\).

• The larger the time constant \(c\), the slower the decay rate of \(e^{-t/c}\), affecting how quickly \(Q(t)\) nears \(a\).
• This relationship is fundamental in electronics, influencing various applications that rely on timing and stability over periods.

Exponential decay provides a predictable pattern that helps in designing systems where charge times are critical, ensuring components like capacitors operate efficiently and effectively.