# 19.1 Electric potential energy: potential difference  (Page 5/10)

 Page 5 / 10

## Electrical potential energy converted to kinetic energy

Calculate the final speed of a free electron accelerated from rest through a potential difference of 100 V. (Assume that this numerical value is accurate to three significant figures.)

Strategy

We have a system with only conservative forces. Assuming the electron is accelerated in a vacuum, and neglecting the gravitational force (we will check on this assumption later), all of the electrical potential energy is converted into kinetic energy. We can identify the initial and final forms of energy to be ${\text{KE}}_{i}=0,\phantom{\rule{0.25em}{0ex}}{\text{KE}}_{f}=½{\mathrm{mv}}^{2},\phantom{\rule{0.25em}{0ex}}{\text{PE}}_{i}=\mathrm{qV}{\text{, and PE}}_{f}=0.$

Solution

Conservation of energy states that

${\text{KE}}_{i}+{\text{PE}}_{i}{\text{= KE}}_{f}+{\text{PE}}_{f}\text{.}$

Entering the forms identified above, we obtain

$\text{qV}=\frac{{\text{mv}}^{2}}{\text{2}}\text{.}$

We solve this for $v$ :

$v=\sqrt{\frac{2\text{qV}}{m}}\text{.}$

Entering values for $q,\phantom{\rule{0.25em}{0ex}}V\text{, and}\phantom{\rule{0.25em}{0ex}}m$ gives

$\begin{array}{lll}v& =& \sqrt{\frac{2\left(–1.60×{\text{10}}^{\text{–19}}\phantom{\rule{0.25em}{0ex}}\text{C}\right)\left(\text{–100 J/C}\right)}{9.11×{\text{10}}^{\text{–31}}\phantom{\rule{0.25em}{0ex}}\text{kg}}}\\ & =& 5.93×{\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}\text{m/s.}\end{array}$

Discussion

Note that both the charge and the initial voltage are negative, as in [link] . From the discussions in Electric Charge and Electric Field , we know that electrostatic forces on small particles are generally very large compared with the gravitational force. The large final speed confirms that the gravitational force is indeed negligible here. The large speed also indicates how easy it is to accelerate electrons with small voltages because of their very small mass. Voltages much higher than the 100 V in this problem are typically used in electron guns. Those higher voltages produce electron speeds so great that relativistic effects must be taken into account. That is why a low voltage is considered (accurately) in this example.

## Section summary

• Electric potential is potential energy per unit charge.
• The potential difference between points A and B, ${V}_{B}–{V}_{A}$ , defined to be the change in potential energy of a charge $q$ moved from A to B, is equal to the change in potential energy divided by the charge, Potential difference is commonly called voltage, represented by the symbol $\text{Δ}V$ .
$\Delta V=\frac{\text{ΔPE}}{q}\phantom{\rule{0.25em}{0ex}}\text{and ΔPE =}\phantom{\rule{0.25em}{0ex}}q\Delta V\text{.}$
• An electron volt is the energy given to a fundamental charge accelerated through a potential difference of 1 V. In equation form,
$\begin{array}{lll}\text{1 eV}& =& \left(1.60×{\text{10}}^{\text{–19}}\phantom{\rule{0.25em}{0ex}}\text{C}\right)\left(1 V\right)=\left(1.60×{\text{10}}^{\text{–19}}\phantom{\rule{0.25em}{0ex}}\text{C}\right)\left(1 J/C\right)\\ & =& 1.60×{\text{10}}^{\text{–19}}\phantom{\rule{0.25em}{0ex}}\text{J.}\end{array}$
• Mechanical energy is the sum of the kinetic energy and potential energy of a system, that is, $\text{KE}+\text{PE}.$ This sum is a constant.

## Conceptual questions

Voltage is the common word for potential difference. Which term is more descriptive, voltage or potential difference?

If the voltage between two points is zero, can a test charge be moved between them with zero net work being done? Can this necessarily be done without exerting a force? Explain.

What is the relationship between voltage and energy? More precisely, what is the relationship between potential difference and electric potential energy?

Voltages are always measured between two points. Why?

How are units of volts and electron volts related? How do they differ?

## Problems&Exercises

Find the ratio of speeds of an electron and a negative hydrogen ion (one having an extra electron) accelerated through the same voltage, assuming non-relativistic final speeds. Take the mass of the hydrogen ion to be $1\text{.}\text{67}×{\text{10}}^{–\text{27}}\phantom{\rule{0.25em}{0ex}}\text{kg}\text{.}$

42.8

#### Questions & Answers

What is conductivity
Saud Reply
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Arjune Reply
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Arjune
derivation of ohms law
Kazeem Reply
derivation of resistance
Kazeem
R=v/I where R=resistor, v=voltage, I=current
Kazeem
magnitude
Arjune
A puck is moving on an air hockey table. Relative to an x, y coordinate system at time t 0 s, the x components of the puck’s ini￾tial velocity and acceleration are v0x 1.0 m/s and ax 2.0 m/s2 . The y components of the puck’s initial velocity and acceleration are v0y 2.0 m/s and ay 2.0
Arjune
Electric current is the flow of electrons
Kelly Reply
is there really flow of electrons exist?
babar
Yes It exists
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babar
if electron flows from where first come and end the first one
babar
an electron will flow accross a conductor because or when it posseses kinectic energy
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electron can not flow jist trasmit electrical energy
ghulam
free electrons of conductor
ankita
electric means the flow heat current.
Serah Reply
electric means the flow of heat current in a circuit.
Serah
What is electric
Manasseh Reply
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ghulam
electric means the flow of heat current in a circuit.
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PhysicswithMrV
d=1.0km÷1000=0.001 t=5×60=300s s=d\t s=0.001/300=0.0000033m\s
Serah
A puck is moving on an air hockey table. Relative to an x, y coordinate system at time t 0 s, the x components of the puck’s ini￾tial velocity and acceleration are v0x 1.0 m/s and ax 2.0 m/s2 . The y components of the puck’s initial velocity and acceleration are v0y 2.0 m/s and ay 2.0
Arjune
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becuse the d .c cannot travel for long distance trnsmission
ghulam
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branch of science which deals with matter energy and their relationship between them
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Life science
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what is heat and temperature
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each of two or more forms of the same element that contain equal numbers of protons but different numbers of neutrons in their nuclei, and hence differ in relative atomic mass but not in chemical properties; in particular, a radioactive form of an element. "some elements have only one stable isotope
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Naveedkhan Reply

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