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We can’t add these forces directly because they don’t point in the same direction: F 12 points only in the − x -direction, while F 13 points only in the + y -direction. The net force is obtained from applying the Pythagorean theorem to its x - and y -components:

F = F x 2 + F y 2

where

F x = F 23 = 1 4 π ε 0 q 2 q 3 r 23 2 = ( 8.99 × 10 9 N · m 2 C 2 ) ( 4.806 × 10 −19 C ) ( 8.01 × 10 −19 C ) ( 4.00 × 10 −7 m ) 2 = 2.16 × 10 −14 N

and

F y = F 21 = 1 4 π ε 0 q 2 q 1 r 21 2 = ( 8.99 × 10 9 N · m 2 C 2 ) ( 4.806 × 10 −19 C ) ( 3.204 × 10 −19 C ) ( 2.00 × 10 −7 m ) 2 = 3.46 × 10 −14 N .

We find that

F = F x 2 + F y 2 = 4.08 × 10 −14 N

at an angle of

ϕ = tan −1 ( F y F x ) = tan −1 ( 3.46 × 10 −14 N −2.16 × 10 −14 N ) = −58 ° ,

that is, 58 ° above the − x -axis, as shown in the diagram.

Significance

Notice that when we substituted the numerical values of the charges, we did not include the negative sign of either q 2 or q 3 . Recall that negative signs on vector quantities indicate a reversal of direction of the vector in question. But for electric forces, the direction of the force is determined by the types (signs) of both interacting charges; we determine the force directions by considering whether the signs of the two charges are the same or are opposite. If you also include negative signs from negative charges when you substitute numbers, you run the risk of mathematically reversing the direction of the force you are calculating. Thus, the safest thing to do is to calculate just the magnitude of the force, using the absolute values of the charges, and determine the directions physically.

It’s also worth noting that the only new concept in this example is how to calculate the electric forces; everything else (getting the net force from its components, breaking the forces into their components, finding the direction of the net force) is the same as force problems you have done earlier.

Check Your Understanding What would be different if q 1 were negative?

The net force would point 58 ° below the − x -axis.

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Summary

  • Coulomb’s law gives the magnitude of the force between point charges. It is
    F 12 ( r ) = 1 4 π ε 0 q 1 q 2 r 12 2 r ^ 12

    where q 2 and q 2 are two point charges separated by a distance r . This Coulomb force is extremely basic, since most charges are due to point-like particles. It is responsible for all electrostatic effects and underlies most macroscopic forces.

Conceptual questions

Would defining the charge on an electron to be positive have any effect on Coulomb’s law?

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An atomic nucleus contains positively charged protons and uncharged neutrons. Since nuclei do stay together, what must we conclude about the forces between these nuclear particles?

The force holding the nucleus together must be greater than the electrostatic repulsive force on the protons.

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Is the force between two fixed charges influenced by the presence of other charges?

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Problems

Two point particles with charges +3 μ C and +5 μ C are held in place by 3-N forces on each charge in appropriate directions. (a) Draw a free-body diagram for each particle. (b) Find the distance between the charges.

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Two charges +3 μ C and +12 μ C are fixed 1 m apart, with the second one to the right. Find the magnitude and direction of the net force on a −2-nC charge when placed at the following locations: (a) halfway between the two (b) half a meter to the left of the +3 μ C charge (c) half a meter above the +12 μ C charge in a direction perpendicular to the line joining the two fixed charges

a. charge 1 is 3 μ C ; charge 2 is 12 μ C , F 31 = 2.16 × 10 −4 N to the left,
F 32 = 8.63 × 10 −4 N to the right,
F net = 6.47 × 10 −4 N to the right;
b. F 31 = 2.16 × 10 −4 N to the right,
F 32 = 9.59 × 10 −5 N to the right,
F net = 3.12 × 10 −4 N to the right,
Three charges are shown. Charge 1 is a 3 micro Coulomb charge at the bottom left. Charge 2 is a 12 micro Coulomb charge at the bottom right, 1 meter to the right of charge 1. Charge 3 is a minus 2 nano Coulomb charge 0.5 meters above charge 2. The charges define a right triangle, with charge 2 at the right angle. The angle at the vertex with charge one is theta. The forces on charge three are shown. F 3 1 points down and to the left, toward charge 1. Force F 3 2 points vertically down. ;
c. F 31 x = −2.76 × 10 −5 N i ^ ,
F 31 y = −1.38 × 10 −5 N j ^ ,
F 32 y = −8.63 × 10 −4 N j ^
F net = −2.76 × 10 −5 N i ^ 8.77 × 10 −4 N j ^

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Questions & Answers

what is calorimeter
odionyenma Reply
heat measuring device
Suvransu
What is mean electric potential
Biren Reply
Electric density formula
Biren Reply
Int E•dA
Vineet
what is relation betweeen potential energy of a system and work done in bringing last chaege from infinity
Shikhar Reply
Einstein and his general theory of relativity, a very nice concept,which revolutionize our modern world.
Antares Reply
what's the theory of relativity?
Piyali
it related to universe time period
jyotirmayee
it is a theory in which time is taken as relative,for diff. motion of objects.
Antares
no !!!
jyotirmayee
it is E=mc'square
jyotirmayee
what is black hole?
jyotirmayee
E=mc^2 is just a reln which sums up everything dude
Antares
what does e=mc^2 stand for?explain
Piyali
m not dude
jyotirmayee
e for speed of light
jyotirmayee
c for speed of light
jyotirmayee
m for mass of object
jyotirmayee
black hole is a super massive object in space gravitation pull of which s so strong dt evn light cannot escape. John wheeler 1st detected it, n concept of it ws given by Einstein himself from his general theory of relativity. Basically v cn si 3 types of it super massive, interstellar n intermediate
Antares
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jyotirmayee
e defined? sorry it,s wrong
jyotirmayee
Why?
Arzoodan
how black holes r formed?
Piyali
e is not speed of light
jyotirmayee
black holes formed when the centre of very massive star collapsed itself
jyotirmayee
e is energy.
Antares
what do u mean by quantum mechanics?
jyotirmayee
in most of d cases black holes r formed by massive collapsed star or star system
Antares
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Antares
exatly...
Arzoodan
Apollo is the name of a satellite !!!!!
jyotirmayee
quantum mechanics is d study of physics describing nature at d smallest level of energy of atoms n subatomic particles.
Antares
qm means explains about the microscopic particles..
KRANTHI
nope name of d sun god
Antares
why the mercury used in thermometers?
jyotirmayee
Ali brother ur exactly spelling is wrong,,,,,
jyotirmayee
why the colour of tube light white?
jyotirmayee
why the mouth became red colour ,,by the regular eating of leaf called ,,betel combining with areca?
jyotirmayee
all formula for calculate specific latent heat of any substance
Idowu Reply
hello
Nigar
Hi, your real Nigga...? Or this jock?
Arzoodan
real
Nigar
You from Pakistan or India?
Arzoodan
c'mon guys.. let's talk Physics.
Yoblaze
mass multiplied with latent heat of a substance
Yoblaze
What you doing about Physics
Arzoodan
you both?
jyotirmayee
in which class?
jyotirmayee
physics is the only subject which underestimated chemistry ,,bio
jyotirmayee
how capacitor is made in inst?
Piyali
what is dipole moment
Bilal Reply
it is the product of electric charges and distance between the two charges
Shikhar
yeap. this correct answer...
Arzoodan
mishra true thanks dia
Ssempala
product of separation of the poles, the rest shikhar got is right
Brad
brad is separation and distance ,are they different?
Ssempala
What is dielectric
Ashis Reply
its a type of medium. generally poor conductors. but their conductivity can be changed
vedanth
you just have to add impurities
vedanth
Thanks
Ssempala
grt
Ssempala
a material which behave as conductor
Shikhar
insulating material, energy level for electron transfer is very high e.g used to increase a magnetic field in a capacitor
Brad
What is the difference between specific heat capacity and heat capacity? Give the equations
elly Reply
presentation on power
Dyutee Reply
relation between Celsius and Kelvin
Anish Reply
0" degree Celsius=273kelvin
jyotirmayee
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
thiru
Its the study of heat and its relation with others kind of energy
Antonio
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
srikanta
@ezra & srikanta; for electrons: a=ke^2/(mr^2) and for protons: a=kp^2/(mr^2)
Sikandar
Practice Key Terms 6

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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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