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Conservation of energy

The total energy of a system is conserved if there is no net addition (or subtraction) due to work or heat transfer. For conservative forces, such as the electrostatic force, conservation of energy states that mechanical energy is a constant.

Mechanical energy is the sum of the kinetic energy and potential energy of a system; that is, K + U = constant . A loss of U for a charged particle becomes an increase in its K . Conservation of energy is stated in equation form as

K + U = constant


K i + U i = K f + U f

where i and f stand for initial and final conditions. As we have found many times before, considering energy can give us insights and facilitate problem solving.

Electrical potential energy converted into 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.)


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 K i = 0 , K f = 1 2 m v 2 , U i = q V , U f = 0 .


Conservation of energy states that

K i + U i = K f + U f .

Entering the forms identified above, we obtain

q V = m v 2 2 .

We solve this for v :

v = 2 q V m .

Entering values for q , V , and m gives

v = 2 ( −1.60 × 10 −19 C ) ( 100 J/C ) 9.11 × 10 −31 kg = 5.93 × 10 6 m/s .


Note that both the charge and the initial voltage are negative, as in [link] . From the discussion of 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. These higher voltages produce electron speeds so great that effects from special relativity must be taken into account and hence are reserved for a later chapter ( Relativity ). That is why we consider a low voltage (accurately) in this example.

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Check Your Understanding How would this example change with a positron? A positron is identical to an electron except the charge is positive.

It would be going in the opposite direction, with no effect on the calculations as presented.

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Voltage and electric field

So far, we have explored the relationship between voltage and energy. Now we want to explore the relationship between voltage and electric field. We will start with the general case for a non-uniform E field. Recall that our general formula for the potential energy of a test charge q at point P relative to reference point R is

U P = R P F · d l .

When we substitute in the definition of electric field ( E = F / q ) , this becomes

U P = q R P E · d l .

Applying our definition of potential ( V = U / q ) to this potential energy, we find that, in general,

Questions & Answers

relation between Celsius and Kelvin
Anish Reply
Newton's second laws is call with
Dyutee Reply
what is mean by thermodynamics
Prasad Reply
it is study about temperature and it's equilibrium
Its the study of heat and its relation with others kind of energy
state caulombs law clearly
constand Reply
show mathematically that an electron has the greater speed than the proton when they attract each other
ezra Reply
show mathematically that an electron has the greater speed than the proton when they attract each other
@ezra & srikanta; for electrons: a=ke^2/(mr^2) and for protons: a=kp^2/(mr^2)
what is electrostatics
Hero Reply
the study of charge at rest
@Hero; the study of charges at rest is the electrostatics
okay what is electrostatic?
charge at rest
set of character...
Gauss law, electric fields, dipoles,...
A proton initially at rest falls through a p.d of 25000V. what speed does it gain?
Minister Reply
@Minister; use equation v= sq root(2×eV/m)
what is the reaction of heat on magnet
Magnetization decreases with increase in temperature. But in case of diamagnetic substance heat has no role on magnetization.
what is a physical significant of electric dipole moment .
A dipole moment it's a mechanical electrical effect used in nature
what is the uses of carbon brushes in generator
Malik Reply
to minimize heat
at what temperature is the degree Fahrenheit equal to degree Celsius
Grace Reply
Celsius and Faharaneith are different, never equal
find their liners express of n=a+b/T² ( plot graph n against T)
Donsmart Reply
Radio Stations often advertis "instant news,,if that meens you can hear the news the instant the radio announcer speaks it is the claim true? what approximate time interval is required for a message to travel from Cairo to Aswan by radio waves (500km) (Assume the waves Casbe detected at this range )
mahmod Reply
what is growth and decay
Pawan Reply
Can someone please predict the trajectory of a point charge in a uniform electric field????
erlinda Reply
Practice Key Terms 4

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Source:  OpenStax, University physics volume 2. OpenStax CNX. Oct 06, 2016 Download for free at http://cnx.org/content/col12074/1.3
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