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Calculating potential of a point charge

Given a point charge q = + 2.0 nC at the origin, calculate the potential difference between point P 1 a distance a = 4.0 cm from q , and P 2 a distance b = 12.0 cm from q , where the two points have an angle of φ = 24 ° between them ( [link] ).

The figure shows two points P subscript 1 and P subscript 2 at distances a and b from the origin and having an angle phi between them.
Find the difference in potential between P 1 and P 2 .

Strategy

Do this in two steps. The first step is to use V B V A = A B E · d l and let A = a = 4.0 cm and B = b = 12.0 cm , with d l = d r = r ^ d r and E = k q r 2 r ^ . Then perform the integral. The second step is to integrate V B V A = A B E · d l around an arc of constant radius r , which means we let d l = r φ ^ d φ with limits 0 φ 24 ° , still using E = k q r 2 r ^ . Then add the two results together.

Solution

For the first part, V B V A = A B E · d l for this system becomes V b V a = a b k q r 2 r ^ · r ^ d r which computes to

Δ V = a b k q r 2 d r = k q [ 1 a 1 b ] = ( 8.99 × 10 9 Nm 2 /C 2 ) ( 2.0 × 10 −9 C ) [ 1 0.040 m 1 0.12 m ] = 300 V .

For the second step, V B V A = A B E · d l becomes Δ V = 0 24 ° k q r 2 r ^ · r φ ^ d φ , but r ^ · φ ^ = 0 and therefore Δ V = 0 . Adding the two parts together, we get 300 V.

Significance

We have demonstrated the use of the integral form of the potential difference to obtain a numerical result. Notice that, in this particular system, we could have also used the formula for the potential due to a point charge at the two points and simply taken the difference.

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Check Your Understanding From the examples, how does the energy of a lightning strike vary with the height of the clouds from the ground? Consider the cloud-ground system to be two parallel plates.

Given a fixed maximum electric field strength, the potential at which a strike occurs increases with increasing height above the ground. Hence, each electron will carry more energy. Determining if there is an effect on the total number of electrons lies in the future.

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Before presenting problems involving electrostatics, we suggest a problem-solving strategy to follow for this topic.

Problem-solving strategy: electrostatics

  1. Examine the situation to determine if static electricity is involved; this may concern separated stationary charges, the forces among them, and the electric fields they create.
  2. Identify the system of interest. This includes noting the number, locations, and types of charges involved.
  3. Identify exactly what needs to be determined in the problem (identify the unknowns). A written list is useful. Determine whether the Coulomb force is to be considered directly—if so, it may be useful to draw a free-body diagram, using electric field lines.
  4. Make a list of what is given or can be inferred from the problem as stated (identify the knowns). It is important to distinguish the Coulomb force F from the electric field E , for example.
  5. Solve the appropriate equation for the quantity to be determined (the unknown) or draw the field lines as requested.
  6. Examine the answer to see if it is reasonable: Does it make sense? Are units correct and the numbers involved reasonable?

Summary

  • Electric potential is potential energy per unit charge.
  • The potential difference between points A and B , V B V A , that is, the change in potential 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 Δ V :
    Δ V = Δ U q or Δ U = q Δ V .
  • An electron-volt is the energy given to a fundamental charge accelerated through a potential difference of 1 V. In equation form,
    1 eV = ( 1.60 × 10 −19 C ) ( 1 V ) = ( 1.60 × 10 −19 C ) ( 1 J/C ) = 1.60 × 10 −19 J .

Questions & Answers

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what is charge
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Mohammed Reply
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Using Kirchhoff's rules, when choosing your loops, can you choose a loop that doesn't have a voltage?
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how was the check your understand 12.7 solved?
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Who is ISSAAC NEWTON
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he's the father of 3 newton law
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he is Chris Issaac's father :)
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what do you understand by drift velocity
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nothing
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well when you apply a small electric field to a conductor that causes to add a little velocity to charged particle than usual, which become their average speed, that is what we call a drift.
graviton
drift velocity
graviton
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It is the amount of other forms of energy converted into electrical energy per unit charge that flow through it.
Brunelle
How electromotive force is differentiated from the terminal voltage?
Danilo
in the emf power is generated while in the terminal pd power is lost.
Brunelle
what is then chemical name of NaCl
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sodium chloride
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sodium chloride
Brunelle
Sodium Chloride.
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common salt
Ahmed
How can we differentiate between static point and test charge?
Comfort Reply
Wat is coplanar in physics
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on the same Cartesian plane
Ahmed
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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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