# 15.1 Ac sources

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By the end of the section, you will be able to:
• Explain the differences between direct current (dc) and alternating current (ac)
• Define characteristic features of alternating current and voltage, such as the amplitude or peak and the frequency

Most examples dealt with so far in this book, particularly those using batteries, have constant-voltage sources. Thus, once the current is established, it is constant. Direct current (dc) is the flow of electric charge in only one direction. It is the steady state of a constant-voltage circuit.

Most well-known applications, however, use a time-varying voltage source. Alternating current (ac) is the flow of electric charge that periodically reverses direction. An ac is produced by an alternating emf, which is generated in a power plant, as described in Induced Electric Fields . If the ac source varies periodically, particularly sinusoidally, the circuit is known as an ac circuit. Examples include the commercial and residential power that serves so many of our needs.

The ac voltages and frequencies commonly used in businesses and homes vary around the world. In a typical house, the potential difference between the two sides of an electrical outlet alternates sinusoidally with a frequency of 60 or 50 Hz and an amplitude of 156 or 311 V, depending on whether you live in the United States or Europe, respectively. Most people know the potential difference for electrical outlets is 120 V or 220 V in the US or Europe, but as explained later in the chapter, these voltages are not the peak values given here but rather are related to the common voltages we see in our electrical outlets. [link] shows graphs of voltage and current versus time for typical dc and ac power in the United States.

Suppose we hook up a resistor to an ac voltage source and determine how the voltage and current vary in time across the resistor. [link] shows a schematic of a simple circuit with an ac voltage source. The voltage fluctuates sinusoidally with time at a fixed frequency, as shown, on either the battery terminals or the resistor. Therefore, the ac voltage    , or the “voltage at a plug,” can be given by

$v={V}_{0}\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\text{ω}t,$

where v is the voltage at time t , ${V}_{0}$ is the peak voltage, and $\text{ω}$ is the angular frequency in radians per second. For a typical house in the United States, ${V}_{0}=156\phantom{\rule{0.2em}{0ex}}\text{V}$ and $\text{ω}=120\pi \phantom{\rule{0.2em}{0ex}}\text{rad/s}$ , whereas in Europe, ${V}_{0}=311\phantom{\rule{0.2em}{0ex}}\text{V}$ and $\text{ω}=100\pi \phantom{\rule{0.2em}{0ex}}\text{rad/s}\text{.}$

For this simple resistance circuit, $I=V/R$ , so the ac current    , meaning the current that fluctuates sinusoidally with time at a fixed frequency, is

$i={I}_{0}\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\text{ω}t,$

where i is the current at time t and ${I}_{0}$ is the peak current and is equal to ${V}_{0}\text{/}R$ . For this example, the voltage and current are said to be in phase, meaning that their sinusoidal functional forms have peaks, troughs, and nodes in the same place. They oscillate in sync with each other, as shown in [link] (b). In these equations, and throughout this chapter, we use lowercase letters (such as i ) to indicate instantaneous values and capital letters (such as I ) to indicate maximum, or peak, values.

Current in the resistor alternates back and forth just like the driving voltage, since $I=V\text{/}R$ . If the resistor is a fluorescent light bulb, for example, it brightens and dims 120 times per second as the current repeatedly goes through zero. A 120-Hz flicker is too rapid for your eyes to detect, but if you wave your hand back and forth between your face and a fluorescent light, you will see the stroboscopic effect of ac.

Check Your Understanding If a European ac voltage source is considered, what is the time difference between the zero crossings on an ac voltage-versus-time graph?

10 ms

## Summary

• Direct current (dc) refers to systems in which the source voltage is constant.
• Alternating current (ac) refers to systems in which the source voltage varies periodically, particularly sinusoidally.
• The voltage source of an ac system puts out a voltage that is calculated from the time, the peak voltage, and the angular frequency.
• In a simple circuit, the current is found by dividing the voltage by the resistance. An ac current is calculated using the peak current (determined by dividing the peak voltage by the resistance), the angular frequency, and the time.

## Conceptual questions

What is the relationship between frequency and angular frequency?

Angular frequency is $2\pi$ times frequency.

## Problems

Write an expression for the output voltage of an ac source that has an amplitude of 12 V and a frequency of 200 Hz.

What is differential form of Gauss's law?
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i have to say. who cares. lol. why know that t all
Jeff
is this just a chat app about the openstax book?
kya ye b.sc ka hai agar haa to konsa part
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it means that the total charge of a body will always be the integral multiples of basic unit charge ( e ) q = ne n : no of electrons or protons e : basic unit charge 1e = 1.602×10^-19
Riya
is the time quantized ? how ?
Mehmet
What do you meanby the statement,"Is the time quantized"
Mayowa
Can you give an explanation.
Mayowa
there are some comment on the time -quantized..
Mehmet
time is integer of the planck time, discrete..
Mehmet
planck time is travel in planck lenght of light..
Mehmet
it's says that charges does not occur in continuous form rather they are integral multiple of the elementary charge of an electron.
Tamoghna
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U can easily calculate work done by 2.303log(v2/v1)
Abhishek
Amount of heat added through q=ncv^delta t
Abhishek
Change in internal energy through q=Q-w
Abhishek
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Prince
g is not gravitacional forcé, is acceleration of gravity of earth and is assumed constante. the "sun g" can not be constant and you should use Newton gravity forcé. by the way its not the "weight" the physical quantity that matters, is the mass
Jorge
Yeah got it... Earth and moon have specific value of g... But in case of sun ☀ it is just a huge sphere of gas...
Prince
Thats why it can't have a constant value of g ....
Prince
not true. you must know Newton gravity Law . even a cloud of gas it has mass thats al matters. and the distsnce from the center of mass of the cloud and the center of the mass of the earth
Jorge
please why is the first law of thermodynamics greater than the second
every law is important, but first law is conservation of energy, this state is the basic in physics, in this case first law is more important than other laws..
Mehmet
First Law describes o energy is changed from one form to another but not destroyed, but that second Law talk about entropy of a system increasing gradually
Mayowa
first law describes not destroyer energy to changed the form, but second law describes the fluid drection that is entropy. in this case first law is more basic accorging to me...
Mehmet
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Phoebe