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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 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
prove that the formula for the unknown resistor is Rx=R2 x R3 divided by R3,when Ig=0.
MAXWELL Reply
what is flux
Bundi Reply
Total number of field lines crossing the surface area
Kamru
Basically flux in general is amount of anything...In Electricity and Magnetism it is the total no..of electric field lines or Magnetic field lines passing normally through the suface
prince
what is temperature change
Celine
a bottle of soft drink was removed from refrigerator and after some time, it was observed that its temperature has increased by 15 degree Celsius, what is the temperature change in degree Fahrenheit and degree Celsius
Celine
process whereby the degree of hotness of a body (or medium) changes
Salim
Q=mcΔT
Salim
where The letter "Q" is the heat transferred in an exchange in calories, "m" is the mass of the substance being heated in grams, "c" is its specific heat capacity and the static value, and "ΔT" is its change in temperature in degrees Celsius to reflect the change in temperature.
Salim
what was the temperature of the soft drink when it was removed ?
Salim
15 degree Celsius
Celine
15 degree
Celine
ok I think is just conversion
Salim
15 degree Celsius to Fahrenheit
Salim
0 degree Celsius = 32 Fahrenheit
Salim
15 degree Celsius = (15×1.8)+32 =59 Fahrenheit
Salim
I dont understand
Celine
the question said you should convert 15 degree Celsius to Fahrenheit
Salim
To convert temperatures in degrees Celsius to Fahrenheit, multiply by 1.8 (or 9/5) and add 32.
Salim
what is d final ans for Fahrenheit and Celsius
Celine
it said what is temperature change in Fahrenheit and Celsius
Celine
the 15 is already in Celsius
Salim
So the final answer for Fahrenheit is 59
Salim
what is d final ans for Fahrenheit and Celsius
Celine
what are the effects of placing a dielectric between the plates of a capacitor
Bundi Reply
increase the capacitance.
Jorge
besides increasing the capacitance, is there any?
Bundi
mechanical stiffness and small size
Jorge
why for an ideal gas internal energy is directly proportional to thermodynamics temperature?
Anne Reply
two charged particles are 8.45cm apart. They are moved and the force on each of them is found to have tripled. How far are they now?
Martin Reply
what is flux
Bundi
Bundi, flux is the number of electric field crossing a surface area
Mubanga
you right
Muhammad
martin,F/F`=(r`×r`)÷(r×r)
Mostafa
determining dimensional correctness
PATRICK Reply
determine dimensional correctness of,T=2π√L/g
PATRICK
somebody help me answer the question above
PATRICK
d=dQ+w
Muhammad
calculate the heat flow per square meter through a mineral roll insulation 5cm thick if the temperature on the two surfaces are 30degree Celsius and 20 degree Celsius respectively. thermal conduction of mineral roll is 0.04
akuribire Reply
what are the elementary compositions of a cell?
jackson Reply
poles, chemical
prabir
when a current pass through a material does the velocity varies
lovet Reply
no.
prabir
what is spin entropy ?and disorder in ferromagnetic material
Nepal Reply
diagram of an hall effect sensor
Aweda Reply
if a magnetised wire having dipole moment M is bent in the form of arc subtending angle of 45°at centre,new magnetic moment is
Pravin Reply
tell the answer
prabir
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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