# 19.2 Electric potential in a uniform electric field

 Page 1 / 5
• Describe the relationship between voltage and electric field.
• Derive an expression for the electric potential and electric field.
• Calculate electric field strength given distance and voltage.

In the previous section, we explored the relationship between voltage and energy. In this section, we will explore the relationship between voltage and electric field. For example, a uniform electric field $\mathbf{\text{E}}$ is produced by placing a potential difference (or voltage) $\Delta V$ across two parallel metal plates, labeled A and B. (See [link] .) Examining this will tell us what voltage is needed to produce a certain electric field strength; it will also reveal a more fundamental relationship between electric potential and electric field. From a physicist’s point of view, either $\Delta V$ or $\mathbf{\text{E}}$ can be used to describe any charge distribution. $\Delta V$ is most closely tied to energy, whereas $\mathbf{\text{E}}$ is most closely related to force. $\Delta V$ is a scalar    quantity and has no direction, while $\mathbf{\text{E}}$ is a vector    quantity, having both magnitude and direction. (Note that the magnitude of the electric field strength, a scalar quantity, is represented by $E$ below.) The relationship between $\Delta V$ and $\mathbf{\text{E}}$ is revealed by calculating the work done by the force in moving a charge from point A to point B. But, as noted in Electric Potential Energy: Potential Difference , this is complex for arbitrary charge distributions, requiring calculus. We therefore look at a uniform electric field as an interesting special case.

The work done by the electric field in [link] to move a positive charge $q$ from A, the positive plate, higher potential, to B, the negative plate, lower potential, is

$W=\phantom{\rule{0.25em}{0ex}}–\Delta \text{PE}=\phantom{\rule{0.25em}{0ex}}–q\Delta V.$

The potential difference between points A and B is

$–\Delta V=\phantom{\rule{0.25em}{0ex}}–\left({V}_{\text{B}}–{V}_{\text{A}}\right)={V}_{\text{A}}–{V}_{\text{B}}={V}_{\text{AB}}.$

Entering this into the expression for work yields

$W={\text{qV}}_{\text{AB}}.$

Work is $W=\text{Fd}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}\theta$ ; here $\text{cos}\phantom{\rule{0.25em}{0ex}}\theta =1$ , since the path is parallel to the field, and so $W=\text{Fd}$ . Since $F=\text{qE}$ , we see that $W=\text{qEd}$ . Substituting this expression for work into the previous equation gives

$\mathrm{qEd}={\text{qV}}_{\text{AB}}.$

The charge cancels, and so the voltage between points A and B is seen to be

$\begin{array}{c}\left(\begin{array}{c}{V}_{\text{AB}}=\mathrm{Ed}\\ E=\frac{{V}_{\text{AB}}}{d}\end{array}}\text{(uniform}\phantom{\rule{0.25em}{0ex}}E\phantom{\rule{0.25em}{0ex}}\text{- field only),}\end{array}$

where $d$ is the distance from A to B, or the distance between the plates in [link] . Note that the above equation implies the units for electric field are volts per meter. We already know the units for electric field are newtons per coulomb; thus the following relation among units is valid:

$\text{1 N}/C=\text{1 V}/m.$

## Voltage between points a and b

$\begin{array}{c}\left(\begin{array}{c}{V}_{\text{AB}}=\mathrm{Ed}\\ E=\frac{{V}_{\text{AB}}}{d}\end{array}}\text{(uniform}\phantom{\rule{0.25em}{0ex}}E\phantom{\rule{0.25em}{0ex}}\text{- field only),}\end{array}$

where $d$ is the distance from A to B, or the distance between the plates.

## What is the highest voltage possible between two plates?

Dry air will support a maximum electric field strength of about $3.0×{\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}\text{V/m}$ . Above that value, the field creates enough ionization in the air to make the air a conductor. This allows a discharge or spark that reduces the field. What, then, is the maximum voltage between two parallel conducting plates separated by 2.5 cm of dry air?

Strategy

We are given the maximum electric field $E$ between the plates and the distance $d$ between them. The equation ${V}_{\text{AB}}=\mathrm{Ed}$ can thus be used to calculate the maximum voltage.

Solution

The potential difference or voltage between the plates is

${\text{V}}_{\text{AB}}=\mathrm{Ed}.$

Entering the given values for $E$ and $d$ gives

${V}_{\text{AB}}=\left(3.0×{\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}\text{V/m}\right)\left(0.025 m\right)=7.5×{\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}V$

or

${V}_{\text{AB}}=\text{75 kV}.$

(The answer is quoted to only two digits, since the maximum field strength is approximate.)

Discussion

One of the implications of this result is that it takes about 75 kV to make a spark jump across a 2.5 cm (1 in.) gap, or 150 kV for a 5 cm spark. This limits the voltages that can exist between conductors, perhaps on a power transmission line. A smaller voltage will cause a spark if there are points on the surface, since points create greater fields than smooth surfaces. Humid air breaks down at a lower field strength, meaning that a smaller voltage will make a spark jump through humid air. The largest voltages can be built up, say with static electricity, on dry days.

Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Got questions? Join the online conversation and get instant answers!