# Voltage

- Page ID
- 248

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)## The Volt

Electrical circuits can be used for sending signals, storing information, or doing calculations, but their most common purpose by far is to manipulate energy, as in the battery-and-bulb example of the previous section. We know that lightbulbs are rated in units of watts, i.e., how many joules per second of energy they can convert into heat and light, but how would this relate to the flow of charge as measured in amperes? By way of analogy, suppose your friend, who didn't take physics, can't find any job better than pitching bales of hay. The number of calories he burns per hour will certainly depend on how many bales he pitches per minute, but it will also be proportional to how much mechanical work he has to do on each bale. If his job is to toss them up into a hayloft, he will got tired a lot more quickly than someone who merely tips bales off a loading dock into trucks. In metric units,

**equation**

Similarly, the rate of energy transformation by a battery will not just depend on how many coulombs per second it pushes through a circuit but also on how much mechanical work it has to do on each coulomb of charge:

**equation**

or

**equation**

Units of joules per coulomb are abbreviated as *volts*, 1 V=1 J/C, named after the Italian physicist Alessandro Volta. Everyone knows that batteries are rated in units of volts, but the voltage concept is more general than that; it turns out that voltage is a property of every point in space.

To gain more insight, let's think again about the analogy with the haybales. It took a certain number of joules of gravitational energy to lift a haybale from one level to another. Since we're talking about gravitational energy, it really makes more sense to talk about units of mass, rather than using the haybale as our measure of the quantity of matter. The gravitational version of voltage would then be joules per kilogram. Gravitational energy equals *mgh*, but if we calculate how much of that we have *per kilogram*, we're canceling out the *m*, giving simply *gh*. For any point in the Earth's gravitational field, we can assign a number, *gh*, which tells us how hard it is to get a given amount of mass to that point. For instance, the top of Mount Everest would have a big value of *gh*, because of the big height. That tells us that it's expensive in terms of energy to lift a given amount of mass from some reference level (sea level, say) to the top of Mount Everest.

Voltage does the same thing, but using electrical energy. We can visualize an electrical circuit as being like a roller-coaster. The battery is like the part of the roller-coaster where they lift you up to the top. The height of this initial hill is analogous to the voltage of the battery. When you roll downhill later, that's like a lightbulb. In the roller-coaster, the initial gravitational energy is turned into heat and sound as the cars go down the hill. In our circuit, the initial electrical energy is turned into heat by the lightbulb (and the hot filament of the lightbulb then glows, turning the heat into light).

### Example 3: Energy stored in a battery

- The 1.2 V rechargeable battery in figure i is labeled 1800 milliamp-hours. What is the maximum amount of energy the battery can store?
- An ampere-hour is a unit of current multiplied by a unit of time. Current is charge per unit time, so an ampere-hour is in fact a funny unit of
*charge*:

**equation**

1800 milliamp-hours is therefore 1800×10^{-3}× 3600 C=6.5×10^{3} C. That's a huge number of charged particles, but the total loss of electrical energy will just be their total charge multiplied by the voltage difference across which they move:

**equation**

Using the definition of voltage, *V*, we can rewrite the equation <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow><mtext>power</mtext><mo lspace="0.278em" rspace="0.278em">=</mo><mtext>current</mtext><mo lspace="0.222em" rspace="0.222em">×</mo><mtext>work per unit charge</mtext></mrow> </math> more concisely as *P*=*IV*.

### Example 4: Units of volt-amps

◊ Doorbells are often rated in volt-amps. What does this combination of units mean?

◊ Current times voltage gives units of power, *P*= *I* *V*, so volt-amps are really just a nonstandard way of writing watts. They are telling you how much power the doorbell requires.

### Example 5: Power dissipated by a battery and bulb

- If a 9.0-volt battery causes 1.0 A to flow through a lightbulb, how much power is dissipated?
- The voltage rating of a battery tells us what voltage difference Δ
*V*it is designed to maintain between its terminals.

**equation**

The only nontrivial thing in this problem was dealing with the units. One quickly gets used to translating common combinations like A⋅*V* into simpler terms.

### Discussion Questions

- In the roller-coaster metaphor, what would a high-voltage roller coaster be like? What would a high-current roller coaster be like?

- Criticize the following statements:
- “He touched the wire, and 10000 volts went through him.”
- “That battery has a charge of 9 volts.”
- “You used up the charge of the battery.”

- When you touch a 9-volt battery to your tongue, both positive and negative ions move through your saliva. Which ions go which way?

- I once touched a piece of physics apparatus that had been wired incorrectly, and got a several-thousand-volt voltage difference across my hand. I was not injured. For what possible reason would the shock have had insufficient power to hurt me?

## Contributors and Attributions

- Benjamin Crowell,
**Conceptual Physics**