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U ( r ) = k q Q r U ref .

A convenient choice of reference that relies on our common sense is that when the two charges are infinitely far apart, there is no interaction between them. (Recall the discussion of reference potential energy in Potential Energy and Conservation of Energy .) Taking the potential energy of this state to be zero removes the term U ref from the equation (just like when we say the ground is zero potential energy in a gravitational potential energy problem), and the potential energy of Q when it is separated from q by a distance r assumes the form

U ( r ) = k q Q r ( z ero reference at r = ) .

This formula is symmetrical with respect to q and Q , so it is best described as the potential energy of the two-charge system.

Potential energy of a charged particle

A + 3.0 -nC charge Q is initially at rest a distance of 10 cm ( r 1 ) from a +5 .0-nC charge q fixed at the origin ( [link] ). Naturally, the Coulomb force accelerates Q away from q , eventually reaching 15 cm ( r 2 ).

The figure shows two positive charges, q (+5.0nC) and Q (+3.0nC) and the repelling force on Q, marked as F subscript e. Q is located at r subscript 1 = 10cm and F subscript e vector is towards r subscript 2 = 15cm.
The charge Q is repelled by q , thus having work done on it and losing potential energy.

What is the change in the potential energy of the two-charge system from r 1 to r 2 ?

Strategy

Calculate the potential energy with the definition given above: Δ U 12 = r 1 r 2 F · d r . Since Q started from rest, this is the same as the kinetic energy.

Solution

We have

Δ U 12 = r 1 r 2 F · d r = r 1 r 2 k q Q r 2 d r = [ k q Q r ] r 1 r 2 = k q Q [ 1 r 2 1 r 1 ] = ( 8.99 × 10 9 Nm 2 /C 2 ) ( 5.0 × 10 −9 C ) ( 3.0 × 10 −9 C ) [ 1 0.15 m 1 0.10 m ] = −4.5 × 10 −7 J .

Significance

The change in the potential energy is negative, as expected, and equal in magnitude to the change in kinetic energy in this system. Recall from [link] that the change in kinetic energy was positive.

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Check Your Understanding What is the potential energy of Q relative to the zero reference at infinity at r 2 in the above example?

It has kinetic energy of 4.5 × 10 −7 J at point r 2 and potential energy of 9.0 × 10 −7 J , which means that as Q approaches infinity, its kinetic energy totals three times the kinetic energy at r 2 , since all of the potential energy gets converted to kinetic.

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Due to Coulomb’s law, the forces due to multiple charges on a test charge Q superimpose; they may be calculated individually and then added. This implies that the work integrals and hence the resulting potential energies exhibit the same behavior. To demonstrate this, we consider an example of assembling a system of four charges.

Assembling four positive charges

Find the amount of work an external agent must do in assembling four charges + 2.0 μ C , + 3.0 μ C , + 4.0 μ C , and + 5.0 μ C at the vertices of a square of side 1.0 cm, starting each charge from infinity ( [link] ).

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.
How much work is needed to assemble this charge configuration?

Strategy

We bring in the charges one at a time, giving them starting locations at infinity and calculating the work to bring them in from infinity to their final location. We do this in order of increasing charge.

Solution

Step 1. First bring the + 2.0 - μ C charge to the origin. Since there are no other charges at a finite distance from this charge yet, no work is done in bringing it from infinity,

W 1 = 0 .

Step 2. While keeping the + 2.0 - μ C charge fixed at the origin, bring the + 3.0 - μ C charge to ( x , y , z ) = ( 1.0 cm , 0 , 0 ) ( [link] ). Now, the applied force must do work against the force exerted by the + 2.0 - μ C charge fixed at the origin. The work done equals the change in the potential energy of the + 3.0 - μ C charge:

Questions & Answers

Figure, shows a pencil partially immersed in a cup of water. Why does the pencil appear to be ?bent
Motion
Abdulrahman
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Sessy Reply
joule
Abdikadir
defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The SI unit of heat capacity is joule per kelvin
Abdikadir
formula of acid hydride
Suman Reply
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Exildah Reply
mass is the amount of staff made up an object
Van
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Abdilahi
mass is accelaration divided force
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mass is force divided by acceleration
Van
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El-shaddai
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Collins Reply
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physics is the study of matter and energy and the interaction between them
Van
physics is the study of energy and matter
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Please canu get more questions on electric field and electric flux please
Gbemisola
is electric field directly proportional to the squared of a distance
Benjamin Reply
No electric field is inversely proportional to the squared distance between the charges
Gbemisola
lets treat linear expansivity please
Ujah Reply
The bullet 2.00cm long is fired at 420/s and passes straight through a 10.0 cm thick board existing at 280 m/s.What is the average acceleration of the bullet through the board?
FAUSTINA
an unstretched spring is 12cm long .A load of 5N stretched it to 15cm .how long will it be under a load of 15N?
Morapeli
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Komolafe
Compare the electric flux through the surface of a cube of side length a that has a charge q at its center to the flux through a spherical surface of radius a with a charge q at its center.
Shari Reply
please I want to know how to solve increase in length
Ujah
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Komolafe
Why a charged capacitor has potential difference but not emf
Gideon Reply
what is the dimension symbol of temperature?
Keren Reply
what is the dimension symbol of temperature?
Keren
what's the meaning of enthalpy in terms of latent heat, internal energy, phase change
Anthony Reply
Enthalpy is the degree ofdisorderlinessof a substance
Gbemisola
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Sangeetha Reply
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Sangeetha
s, p, d, f in this table
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Vengata
Periodic table
Gbemisola
what are waves
KENNETH Reply
In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities
Abdikadir
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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