### Learning Objectives

By the finish of this section, you will be able to do the following:

State Coulomb"s legislation in state of just how the electrostatic force transforms with the distance in between two objectsCalculate the electrostatic force in between two suggest charges, such together electrons or protonsCompare the electrostatic force to the gravitational attraction because that a proton and also an electron; for a human and also Earth**3.A.3.3**The college student is able to describe a pressure as an interaction between two objects and identify both objects for any kind of force.

**(S.P. 1.4)**

**3.A.3.4**The college student is may be to make claims around the force on an object due come the visibility of other objects through the exact same property: mass, electric charge.

**(S.P. 6.1, 6.4)**

**3.C.2.1**The college student is may be to usage Coulomb"s legislation qualitatively and quantitatively to do predictions around the interaction between two electric suggest charges—interactions between collections the electric point charges space not covered in Physics 1 and instead are minimal to Physics 2.

**(S.P. 2.2, 6.4)**

**3.C.2.2**The student is able to affix the principles of gravitational force and also electric force to compare similarities and also differences between the forces.

**(S.P. 7.2)**

Figure 1.18 This NASA picture of Arp 87 mirrors the an outcome of a strong gravitational attraction in between two galaxies. In contrast, at the subatomic level, the electrostatic attraction in between two objects, such as an electron and also a proton, is much greater 보다 their common attraction because of gravity. (NASA/HST)

Through the work-related of researchers in the late eighteenth century, the main attributes of the electrostatic force—the existence of two varieties of charge, the monitoring that choose charges repel, unequal charges attract, and also the diminish of force with distance—were ultimately refined, and also expressed together a mathematics formula. The mathematics formula for the electrostatic pressure is dubbed Coulomb"s legislation after the French physicist Charles Coulomb (1736–1806), that performed experiments and first proposed a formula to calculate it.

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1.3 F=k|q1q2|r2F=k|q1q2|r2 size 12F=k q rSub dimension 81 q rSub size 82 over r rSup size 82

Coulomb"s law calculates the size of the pressure FF in between two allude charges, q1q1 size 12q rSub dimension 81 and q2q2 dimension 12q rSub size 82 , separated by a distance rr. In SI units, the consistent kk is equal to

1.4 k=8.988×109N⋅m2C2≈8.99×109N⋅m2C2.k=8.988×109N⋅m2C2≈8.99×109N⋅m2C2. Dimension 12k=8 "." "988" times "10" rSup dimension 89 N cdot m rSup dimension 82 end C rSup dimension 82 approx 9 "." "00" time "10" rSup size 89 N cdot m rSup dimension 82 end C rSup dimension 82

The electrostatic pressure is a vector quantity and is expressed in units of newtons. The force is interpreted to be follow me the line joining the two charges (see number 1.19).

Although the formula because that Coulomb"s law is simple, it to be no mean task to prove it. The experiments Coulomb did, through the primitive devices then available, to be difficult. Modern experiments have actually verified Coulomb"s regulation to great precision. Because that example, it has been displayed that the pressure is inversely proportional come distance between two objects squared F∝1/r2F∝1/r2 dimension 12 left (F prop 1 slash r rSup size 82 best ) come an accuracy of 1 part in 1016.1016. Dimension 12"10" rSup dimension 8"16" No exception have ever before been found, also at the small distances within the atom.

Figure 1.19 The size of the electrostatic pressure FF dimension 12F between allude charges q1q1 dimension 12q rSub size 81 and also q2q2 dimension 12q rSub size 82 be separated by a distance rr dimension 12F is given by Coulomb"s law. Keep in mind that Newton"s third law—every force exerted create an equal and also opposite force—applies together usual—the pressure on q1q1 size 12q rSub size 81 is same in magnitude and also opposite in direction to the force it exerts ~ above q2q2 size 12q rSub size 82 . (a) like charges. (b) unlike charges.

Recall the the gravitational force (Newton"s law of gravitation) quantifies force as Fs=GmMr2Fs=GmMr2.

The comparison between the 2 forces—gravitational and also electrostatic—shows some similarities and differences. Gravitational force is proportional to the masses of connecting objects, and the electrostatic force is proportional to the magnitudes that the dues of connecting objects. For this reason both forces are proportional come a residential or commercial property that represents the stamin of interaction for a given field. In addition, both pressures are inversely proportional to the square of the distances between them. It might seem that the two pressures are related but that is no the case. In fact, over there are substantial variations in the magnitudes the the two forces as they depend on various parameters and different mechanisms. Because that electrons (or protons), electrostatic pressure is dominant and also is much greater than the gravitational force. On the various other hand, gravitational force is generally leading for objects with large masses. Another significant difference in between the two pressures is that gravitational pressure can just be attractive, whereas electrostatic could be attractive or repulsive, depending upon the authorize of charges; unequal charges attract and also like dues repel.

### Example 1.1 How strong is the Coulomb pressure Relative to the Gravitational Force?

Compare the electrostatic force between an electron and also proton separated by 0.530×10−10m0.530×10−10m dimension 120 "." "530" times "10" rSup size 8 - "10" m v the gravitational force in between them. This street is their mean separation in a hydrogen atom.

**Strategy**

To to compare the two forces, we very first compute the electrostatic pressure using Coulomb"s law, F=k|q1q2|r2F=k|q1q2|r2 dimension 12F=k q rSub dimension 81 q rSub size 82 end r rSup size 82 . Us then calculate the gravitational pressure using Newton"s universal legislation of gravitation. Finally, we take a ratio to see exactly how the pressures compare in magnitude.

**Solution**

Entering the given and also known information about the charges and separation that the electron and also proton into the expression that Coulomb"s regulation yields

1.5 F=k|q1q2|r2F=k|q1q2|r2 size 12F=k q rSub size 81 q rSub dimension 82 end r rSup size 82

1.6 =8.99× 109N⋅ m2/C2×(1.60× 10–19C)(1.60× 10–19 C)(0.530× 10–10m)2.=8.99× 109N⋅ m2/C2×(1.60× 10–19C)(1.60× 10–19 C)(0.530× 10–10m)2.alignl stack size 12" "= left (9 "." "00 " times " 10" rSup size 89 N cdot " m" rSup size 82 /C rSup dimension 82 ideal ) time ( "-1" "." "60 " times " 10" rSup dimension 8"-19" C ) ( 1 "." "60" time " 10" rSup dimension 8"-19 " C ) over ( 0 "." "530 " times " 10" rSup dimension 8"-10" m ) rSup size 82 #

The charges are opposite in sign, for this reason this is one attractive force. This is a very huge force for an electron—it would cause an acceleration that 8.99×1022m/s28.99×1022m/s2 size 129 "." "00" time "10" rSup size 8"22" m slash s rSup size 82 , verification is left together an end-of-section problem.The gravitational force is offered by Newton"s legislation of gravitation as

1.8 FG= G mMr2,FG= G mMr2, dimension 12F rSub dimension 8G =" G " "mM" over r rSup dimension 82

where G=6.67×10−11N⋅m2/kg2G=6.67×10−11N⋅m2/kg2 dimension 12G=6 "." "67" time "10" rSup size 8 - "11" N cdot m rSup dimension 82 cut ital "kg" rSup dimension 82 . Right here mm and also MM stand for the electron and also proton masses, which deserve to be uncovered in the appendices. Entering worths for the knowns yields

1.9 FG=(6.67× 10–11N⋅m2/kg2)×(9.11× 10–31kg)(1.67×10–27kg)(0.530× 10–10m)2=3.61× 10–47N.FG=(6.67× 10–11N⋅m2/kg2)×(9.11× 10–31kg)(1.67×10–27kg)(0.530× 10–10m)2=3.61× 10–47N.

This is additionally an attractive force, although it is traditionally shown as positive because gravitational pressure is always attractive. The proportion of the magnitude of the electrostatic pressure to gravitational pressure in this situation is, thus

1.10 FFG= 2.27 × 1039.FFG= 2.27 × 1039. Size 12 F end F rSub size 8G =" 2" "." "27 " time " 10" rSup dimension 8"39"

**Discussion**

This is a remarkably large ratio! keep in mind that this will be the ratio of electrostatic force to gravitational force for an electron and also a proton at any distance, acquisition the ratio prior to entering numerical values mirrors that the street cancels. This ratio provides some point out of just how much bigger the Coulomb pressure is than the gravitational force in between two that the most typical particles in nature.

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As the instance implies, gravitational force is totally negligible on a little scale, wherein the interaction of individual fee particles are important. ~ above a big scale, such as in between Earth and also a person, the reverse is true. Most objects are almost electrically neutral, and also so attractive and also repulsive Coulomb forces practically cancel. Gravitational force on a big scale dominates interaction between big objects since it is constantly attractive, when Coulomb pressures tend come cancel.