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W 2 = k q 1 q 2 r 12 = ( 9.0 × 10 9 N · m 2 C 2 ) ( 2.0 × 10 −6 C ) ( 3.0 × 10 −6 C ) 1.0 × 10 −2 m = 5.4 J .
The figure shows a square with side length 1.0cm and two charges (2.0µC and 3.0µC) on adjacent corners.
Step 2. Work W 2 to bring the + 3.0 - μ C charge from infinity.

Step 3. While keeping the charges of + 2.0 μ C and + 3.0 μ C fixed in their places, bring in the + 4.0 - μ C charge to ( x , y , z ) = ( 1.0 cm , 1.0 cm , 0 ) ( [link] ). The work done in this step is

W 3 = k q 1 q 3 r 13 + k q 2 q 3 r 23 = ( 9.0 × 10 9 N · m 2 C 2 ) [ ( 2.0 × 10 −6 C ) ( 4.0 × 10 −6 C ) 2 × 10 −2 m + ( 3.0 × 10 −6 C ) ( 4.0 × 10 −6 C ) 1.0 × 10 −2 m ] = 15.9 J .
The figure shows a square with side length 1.0cm and three charges (2.0µC, 3.0µC and 4.0µC) on three corners.
Step 3. The work W 3 to bring the + 4.0 - μ C charge from infinity.

Step 4. Finally, while keeping the first three charges in their places, bring the + 5.0 - μ C charge to ( x , y , z ) = ( 0 , 1.0 cm , 0 ) ( [link] ). The work done here is

W 4 = k q 4 [ q 1 r 14 + q 2 r 24 + q 3 r 34 ] , = ( 9.0 × 10 9 N · m 2 C 2 ) ( 5.0 × 10 −6 C ) [ ( 2.0 × 10 −6 C ) 1.0 × 10 −2 m + ( 3.0 × 10 −6 C ) 2 × 10 −2 m + ( 4.0 × 10 −6 C ) 1.0 × 10 −2 m ] = 36.5 J .
The figure shows a square with side length 1.0cm and four charges (2.0µC, 3.0µC, 4.0µC and 5.0µC) located at four corners.
Step 4. The work W 4 to bring the + 5.0 - μ C charge from infinity.

Hence, the total work done by the applied force in assembling the four charges is equal to the sum of the work in bringing each charge from infinity to its final position:

W T = W 1 + W 2 + W 3 + W 4 = 0 + 5.4 J + 15.9 J + 36.5 J = 57.8 J .

Significance

The work on each charge depends only on its pairwise interactions with the other charges. No more complicated interactions need to be considered; the work on the third charge only depends on its interaction with the first and second charges, the interaction between the first and second charge does not affect the third.

Check Your Understanding Is the electrical potential energy of two point charges positive or negative if the charges are of the same sign? Opposite signs? How does this relate to the work necessary to bring the charges into proximity from infinity?

positive, negative, and these quantities are the same as the work you would need to do to bring the charges in from infinity

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Note that the electrical potential energy is positive if the two charges are of the same type, either positive or negative, and negative if the two charges are of opposite types. This makes sense if you think of the change in the potential energy Δ U as you bring the two charges closer or move them farther apart. Depending on the relative types of charges, you may have to work on the system or the system would do work on you, that is, your work is either positive or negative. If you have to do positive work on the system (actually push the charges closer), then the energy of the system should increase. If you bring two positive charges or two negative charges closer, you have to do positive work on the system, which raises their potential energy. Since potential energy is proportional to 1/ r , the potential energy goes up when r goes down between two positive or two negative charges.

On the other hand, if you bring a positive and a negative charge nearer, you have to do negative work on the system (the charges are pulling you), which means that you take energy away from the system. This reduces the potential energy. Since potential energy is negative in the case of a positive and a negative charge pair, the increase in 1/ r makes the potential energy more negative, which is the same as a reduction in potential energy.

The result from [link] may be extended to systems with any arbitrary number of charges. In this case, it is most convenient to write the formula as

W 12 N = k 2 i N j N q i q j r i j for i j .

The factor of 1/2 accounts for adding each pair of charges twice.

Summary

  • The work done to move a charge from point A to B in an electric field is path independent, and the work around a closed path is zero. Therefore, the electric field and electric force are conservative.
  • We can define an electric potential energy, which between point charges is U ( r ) = k q Q r , with the zero reference taken to be at infinity.
  • The superposition principle holds for electric potential energy; the potential energy of a system of multiple charges is the sum of the potential energies of the individual pairs.

Conceptual questions

Would electric potential energy be meaningful if the electric field were not conservative?

No. We can only define potential energies for conservative fields.

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Why do we need to be careful about work done on the system versus work done by the system in calculations?

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Does the order in which we assemble a system of point charges affect the total work done?

No, though certain orderings may be simpler to compute.

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Problems

Consider a charge Q 1 ( + 5.0 μ C ) fixed at a site with another charge Q 2 (charge + 3.0 μ C , mass 6.0 μ g ) moving in the neighboring space. (a) Evaluate the potential energy of Q 2 when it is 4.0 cm from Q 1 . (b) If Q 2 starts from rest from a point 4.0 cm from Q 1 , what will be its speed when it is 8.0 cm from Q 1 ? ( Note: Q 1 is held fixed in its place.)

a. U = 3.4 J;
b. 1 2 m v 2 = k Q 1 Q 2 ( 1 r i 1 r f ) v = 750 m/s

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Two charges Q 1 ( + 2.00 μ C ) and Q 2 ( + 2.00 μ C ) are placed symmetrically along the x -axis at x = ± 3.00 cm . Consider a charge Q 3 of charge + 4.00 μ C and mass 10.0 mg moving along the y -axis. If Q 3 starts from rest at y = 2.00 cm, what is its speed when it reaches y = 4.00 cm?

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To form a hydrogen atom, a proton is fixed at a point and an electron is brought from far away to a distance of 0.529 × 10 −10 m, the average distance between proton and electron in a hydrogen atom. How much work is done?

U = 4.36 × 10 −18 J

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(a) What is the average power output of a heart defibrillator that dissipates 400 J of energy in 10.0 ms? (b) Considering the high-power output, why doesn’t the defibrillator produce serious burns?

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Questions & Answers

it is the work done in moving a charge to a point from infinity against electric field
Ashok Reply
what is the weight of the earth in space
peterpaul Reply
As w=mg where m is mass and g is gravitational force... Now if we consider the earth is in gravitational pull of sun we have to use the value of "g" of sun, so we can find the weight of eaeth in sun with reference to sun...
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
Ifeoma Reply
define electric image.obtain expression for electric intensity at any point on earthed conducting infinite plane due to a point charge Q placed at a distance D from it.
Mateshwar Reply
explain the lack of symmetry in the field of the parallel capacitor
Phoebe Reply
pls. explain the lack of symmetry in the field of the parallel capacitor
Phoebe
does your app come with video lessons?
Ahmed Reply
What is vector
Ajibola Reply
Vector is a quantity having a direction as well as magnitude
Damilare
tell me about charging and discharging of capacitors
Ahemen Reply
a big and a small metal spheres are connected by a wire, which of this has the maximum electric potential on the surface.
Bundi Reply
3 capacitors 2nf,3nf,4nf are connected in parallel... what is the equivalent capacitance...and what is the potential difference across each capacitor if the EMF is 500v
Prince Reply
equivalent capacitance is 9nf nd pd across each capacitor is 500v
santanu
four effect of heat on substances
Prince Reply
why we can find a electric mirror image only in a infinite conducting....why not in finite conducting plate..?
Rima Reply
because you can't fit the boundary conditions.
Jorge
what is the dimensions for VISCOUNSITY (U)
Branda
what is thermodynamics
Aniket Reply
the study of heat an other form of energy.
John
heat is internal kinetic energy of a body but it doesnt mean heat is energy contained in a body because heat means transfer of energy due to difference in temperature...and in thermo-dynamics we study cause, effect, application, laws, hypothesis and so on about above mentioned phenomenon in detail.
ing
It is abranch of physical chemistry which deals with the interconversion of all form of energy
Vishal
what is colamb,s law.?
Muhammad Reply
it is a low studied the force between 2 charges F=q.q`\r.r
Mostafa
what is the formula of del in cylindrical, polar media
Birengeso Reply
Practice Key Terms 1

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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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