DC Circuits Containing Resistors and Capacitors
A detailed discussion of how to make precise measurements is not only important for practical applications. It is also a good way to get acquainted with the tremendous predictive power that our physical laws provide. This understanding helps us design devices to reveal what is going on inside a circuit even though it is all happening at a microscopic level, invisible to the naked eye. Measurement devices are the bridge between this micro-world and the macroscopic world of pointer arrows and displays that we can perceive and interpret.
The range of validity of Kirchhoff's rules is broad. Simply put, Kirchhoff's rules are valid whenever the conservation laws on which they are based are valid. The following three energy forms are accounted for in Kirchhoff's rules: electrical potential energy, the energy generating an electromotive force, and the energy lost to heat in a resistor. An energy form that is not included in Kirchoff's rules is electromagnetic radiation.
With this in mind, we can apply Kirchhoff's Laws to many other types of circuits, containing not just batteries and resistors. In this we add capacitors to the mix. As you recall from our discussion of capacitance, a capacitor is able to store electric charge in an amount proportional to the voltage between its plates.
Read this text to explore a topic we have not yet discussed: the details of the charging and discharging process. Since this process involves current flow, we have to look at capacitors in a complete circuit.
When you use a flash camera, it takes a few seconds to charge the capacitor that powers the flash. The light flash discharges the capacitor in a tiny fraction of a second. Why does charging take longer than discharging? This question and a number of other phenomena that involve charging and discharging capacitors are discussed in this module.
RC Circuits
An RC circuit is one containing a resistor R and a capacitor C. The capacitor is an electrical component that stores electric charge.
Figure 21.38 shows a simple RC circuit that employs a DC (direct current) voltage source. The capacitor is initially uncharged. As soon as the switch is closed, current flows to and from the initially uncharged capacitor. As charge increases on the capacitor plates, there is increasing opposition to the flow of charge by the repulsion of like charges on each plate.
In terms of voltage, this is because voltage across the capacitor is given by , where is the amount of charge stored on each plate and is the capacitance. This voltage opposes the battery, growing from zero to the maximum emf when fully charged. The current thus decreases from its initial value of to zero as the voltage on the capacitor reaches the same value as the emf. When there is no current, there is no drop, and so the voltage on the capacitor must then equal the emf of the voltage source. This can also be explained with Kirchhoff’s second rule (the loop rule), discussed in Kirchhoff’s Rules, which says that the algebraic sum of changes in potential around any closed loop must be zero.
The initial current is , because all of the drop is in the resistance. Therefore, the smaller the resistance, the faster a given capacitor will be charged. Note that the internal resistance of the voltage source is included
in , as are the resistances of the capacitor and the connecting wires. In the flash camera scenario above, when the batteries powering the camera begin to wear out, their internal resistance rises, reducing the current and lengthening the time
it takes to get ready for the next flash.
Figure 21.38 (a) An circuit with an initially uncharged capacitor. Current flows in the direction shown (opposite of electron flow) as soon as the switch is closed. Mutual repulsion of like charges in the capacitor progressively slows the flow as the capacitor is charged, stopping the current when the capacitor is fully charged and . (b) A graph of voltage across the capacitor versus time, with the switch closing at time . (Note that in the two parts of the figure, the capital script E stands for emf, stands for the charge stored on the capacitor, and is the time constant.)
Voltage on the capacitor is initially zero and rises rapidly at first, since the initial current is a maximum. Figure 21.38(b) shows a graph of capacitor voltage versus time () starting when the switch is closed at . The
voltage approaches emf asymptotically, since the closer it gets to emf the less current flows. The equation for voltage versus time when charging a capacitor through a resistor , derived using calculus, is
where is the voltage across the capacitor, emf is equal to the emf of the DC voltage source, and the exponential e = 2.718 … is the base of the natural logarithm. Note that the units of are seconds. We define
where (the Greek letter tau) is called the time constant for an circuit. As noted before, a small resistance allows the capacitor to charge faster. This is reasonable, since a larger current flows through a smaller resistance. It is also reasonable that the smaller the capacitor , the less time needed to charge it. Both factors are contained in .
More quantitatively, consider what happens when . Then the voltage on the capacitor is
This means that in the time , the voltage rises to 0.632 of its final value. The voltage will rise 0.632 of the remainder in the next time . It is a characteristic of the exponential function that the final value is never reached, but 0.632 of the remainder to that value is achieved in every time, . In just a few multiples of the time constant , then, the final value is very nearly achieved, as the graph in Figure 21.38(b) illustrates.
Discharging a Capacitor
Discharging a capacitor through a resistor proceeds in a similar fashion, as Figure 21.39 illustrates. Initially, the current is , driven by the initial voltage on the capacitor. As the voltage decreases, the current and hence the rate of discharge decreases, implying another exponential formula for . Using calculus, the voltage on a capacitor being discharged through a resistor is found to be
Figure 21.39 (a) Closing the switch discharges the capacitor through the resistor . Mutual repulsion of like charges on each plate drives the current. (b) A graph of voltage across the capacitor versus time, with at . The voltage decreases exponentially, falling a fixed fraction of the way to zero in each subsequent time constant .
The graph in Figure 21.39 (b) is an example of this exponential decay. Again, the time constant is . A small resistance allows the capacitor to discharge in a small time, since the current is larger. Similarly,
a small capacitance requires less time to discharge, since less charge is stored. In the first time interval after the switch is closed, the voltage falls to 0.368 of its initial value, since .
During each successive time , the voltage falls to 0.368 of its preceding value. In a few multiples of , the voltage becomes very close to zero, as indicated by the graph in Figure 21.39(b).
Now we can explain why the flash camera in our scenario takes so much longer to charge than discharge; the resistance while charging is significantly greater than while discharging. The internal resistance of the battery accounts for most of the resistance while charging. As the battery ages, the increasing internal resistance makes the charging process even slower. (You may have noticed this.)
The flash discharge is through a low-resistance ionized gas in the flash tube and proceeds very rapidly. Flash photographs, such as in Figure 21.40, can capture a brief instant of a rapid motion because the flash can be less than a microsecond in duration. Such flashes can be made extremely intense.
During World War II, nighttime reconnaissance photographs were made from the air with a single flash illuminating more than a square kilometer of enemy territory. The brevity of the flash eliminated blurring due to the surveillance aircraft’s motion.
Today, an important use of intense flash lamps is to pump energy into a laser. The short intense flash can rapidly energize a laser and allow it to reemit the energy in another form.
Figure 21.40 This stop-motion photograph of a rufous hummingbird (Selasphorus rufus) feeding on a flower was obtained with an extremely brief and intense flash of light powered by the discharge of a capacitor through a gas. (credit: Dean E. Biggins, U.S. Fish and Wildlife Service)
Example 21.6 Integrated Concept Problem: Calculating Capacitor Size – Strobe Lights
High-speed flash photography was pioneered by Doc Edgerton in the 1930s, while he was a professor of electrical engineering at MIT. You might have seen examples of his work in the amazing shots of hummingbirds in motion, a drop of milk splattering on a table, or a bullet penetrating an apple (see Figure 21.40). To stop the motion and capture these pictures, one needs a high-intensity, very short pulsed flash, as mentioned earlier in this module.
Suppose one wished to capture the picture of a bullet (moving at ) that was passing through an apple. The duration of the flash is related to the time constant, . What size capacitor would one need in the circuit to succeed, if the resistance of the flash tube was ? Assume the apple is a sphere with a diameter of .
Strategy
We begin by identifying the physical principles involved. This example deals with the strobe light, as discussed above. Figure 21.39 shows the circuit for this probe. The characteristic time of the strobe is given as .
Solution
We wish to find , but we don’t know . We want the flash to be on only while the bullet traverses the apple. So we need to use the kinematic equations that describe the relationship between distance , velocity , and time :
The bullet’s velocity is given as , and the distance is . The traverse time, then, is
We set this value for the crossing time $$t$ equal to . Therefore,
(Note: Capacitance is typically measured in farads, , defined as Coulombs per volt. From the equation, we see that can also be stated in units of seconds per ohm.)
Discussion
The flash interval of (the traverse time of the bullet) is relatively easy to obtain today. Strobe lights have opened up new worlds from science to entertainment. The information from the picture of the apple and bullet was used in the Warren Commission Report on the assassination of President John F. Kennedy in 1963 to confirm that only one bullet was fired.
RC Circuits for Timing
circuits are commonly used for timing purposes. A mundane example of this is found in the ubiquitous intermittent wiper systems of modern cars. The time between wipes is varied by adjusting the resistance in an circuit. Another example of an circuit is found in novelty jewelry, Halloween costumes, and various toys that have battery-powered flashing lights. (See Figure 21.41 for a timing circuit.)
A more crucial use of circuits for timing purposes is in the artificial pacemaker, used to control heart rate. The heart rate is normally controlled by electrical signals generated by the sino-atrial (SA) node, which is on the wall of the right atrium chamber. This causes the muscles to contract and pump blood. Sometimes the heart rhythm is abnormal and the heartbeat is too high or too low.
The artificial pacemaker is inserted near the heart to provide electrical signals to the heart when needed with the appropriate time constant. Pacemakers have sensors that detect body motion and breathing to increase the heart rate during exercise to
meet the body’s increased needs for blood and oxygen.
Figure 21.41 (a) The lamp in this circuit ordinarily has a very high resistance, so that the battery charges the capacitor as if the lamp were not there. When the voltage reaches a threshold value, a current flows through the lamp that dramatically reduces its resistance, and the capacitor discharges through the lamp as if the battery and charging resistor were not there. Once discharged, the process starts again, with the flash period determined by the constant . (b) A graph of voltage versus time for this circuit.
Example 21.7 Calculating Time: RC Circuit in a Heart Defibrillator
A heart defibrillator is used to resuscitate an accident victim by discharging a capacitor through the trunk of her body. A simplified version of the circuit is seen in Figure 21.39. (a) What is the time constant if an capacitor is used and the path resistance through her body is ? (b) If the initial voltage is 10.0 kV, how long does it take to decline to ?
Strategy
Since the resistance and capacitance are given, it is straightforward to multiply them to give the time constant asked for in part (a). To find the time for the voltage to decline to , we repeatedly multiply the initial voltage by 0.368 until a voltage less than or equal to is obtained. Each multiplication corresponds to a time of seconds.
Solution for (a)
The time constant is given by the equation . Entering the given values for resistance and capacitance (and remembering that units for a farad can be expressed as ) gives
Solution for (b)
In the first 8.00 ms, the voltage (10.0 kV) declines to 0.368 of its initial value. That is:
(Notice that we carry an extra digit for each intermediate calculation.) After another 8.00 ms, we multiply by 0.368 again, and the voltage is
Similarly, after another 8.00 ms, the voltage is
Discussion
So after only 24.0 ms, the voltage is down to 498 V, or 4.98% of its original value.Such brief times are useful in heart defibrillation, because the brief but intense current causes a brief but effective contraction of the heart. The actual circuit in a heart defibrillator is slightly more complex than the one in Figure 21.39, to compensate for magnetic and AC effects that will be covered in Magnetism.
Source: Rice University, https://openstax.org/books/college-physics/pages/21-6-dc-circuits-containing-resistors-and-capacitors
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