By the end of this section, you will be able to:
Through the work of scientists in the late 18th century, the main features of the electrostatic force—the existence of two types of charge, the observation that like charges repel, unlike charges attract, and the decrease of force with distance—were eventually refined, and expressed as a mathematical formula. The mathematical formula for the electrostatic force is called Coulomb’s law after the French physicist Charles Coulomb (1736–1806), who performed experiments and first proposed a formula to calculate it.
[latex]\displaystyle{F}=k\frac{\mid{q}_1q_2\mid}{r^2}\\[/latex] Coulomb’s law calculates the magnitude of the force F between two point charges, q1 and q2, separated by a distance r. In SI units, the constant k is equal to [latex]\displaystyle{k}=8.988\times10^9\frac{\text{N}\cdot\text{m}^2}{\text{C}^2}\approx8.99\times10^9\frac{\text{N}\cdot\text{m}^2}{\text{C}^2}\\[/latex] The electrostatic force is a vector quantity and is expressed in units of newtons. The force is understood to be along the line joining the two charges. (See Figure 2). Although the formula for Coulomb’s law is simple, it was no mean task to prove it. The experiments Coulomb did, with the primitive equipment then available, were difficult. Modern experiments have verified Coulomb’s law to great precision. For example, it has been shown that the force is inversely proportional to distance between two objects squared [latex]\left(F\propto\frac{1}{r^2}\right)\\[/latex] to an accuracy of 1 part in 1016. No exceptions have ever been found, even at the small distances within the atom.
Compare the electrostatic force between an electron and proton separated by 0.530 × 10−10 m with the gravitational force between them. This distance is their average separation in a hydrogen atom. StrategyTo compare the two forces, we first compute the electrostatic force using Coulomb’s law, [latex]\displaystyle{F}=k\frac{\mid{q}_1q_2\mid}{r^2}\\[/latex]. We then calculate the gravitational force using Newton’s universal law of gravitation. Finally, we take a ratio to see how the forces compare in magnitude. SolutionEntering the given and known information about the charges and separation of the electron and proton into the expression of Coulomb’s law yields [latex]\begin{array}{lll}{F}&=&k\frac{\mid{q}_1q_2\mid}{r^2}\\\text{ }&=&8.99\times10^9\frac{\text{N}\cdot\text{m}^2}{\text{C}^2}\times\frac{\left(1.60\times10^{-19}\text{ C}\right)\left(1.60\times10^{-19}\text{ C}\right)}{\left(0.530\times10^{-10}\text{ m}\right)^2}\end{array}\\[/latex] Thus the Coulomb force is F = 8.19 × 10−8 N. The charges are opposite in sign, so this is an attractive force. This is a very large force for an electron—it would cause an acceleration of 8.99 × 1022 m/s2 (verification is left as an end-of-section problem).The gravitational force is given by Newton’s law of gravitation as: [latex]F_G=G\frac{mM}{r^2}\\[/latex], where G = 6.67 × 10−11 N · m2/kg2. Here m and M represent the electron and proton masses, which can be found in the appendices. Entering values for the knowns yields [latex]F_G=\left(6.67\times10^{-11}\text{N}\cdot\text{m}^2\text{/kg}^2\right)\times\frac{\left(9.11\times10^{-31}\text{ kg}\right)\left(1.67\times10^{-27}\text{ kg}\right)}{\left(0.530\times10^{-10}\text{ m}\right)^2}=3.61\times10^{-47}\text{ N}\\[/latex] This is also an attractive force, although it is traditionally shown as positive since gravitational force is always attractive. The ratio of the magnitude of the electrostatic force to gravitational force in this case is, thus, [latex]\frac{F}{F_G}=2.27\times10^{39}\\[/latex]. DiscussionThis is a remarkably large ratio! Note that this will be the ratio of electrostatic force to gravitational force for an electron and a proton at any distance (taking the ratio before entering numerical values shows that the distance cancels). This ratio gives some indication of just how much larger the Coulomb force is than the gravitational force between two of the most common particles in nature. As the example implies, gravitational force is completely negligible on a small scale, where the interactions of individual charged particles are important. On a large scale, such as between the Earth and a person, the reverse is true. Most objects are nearly electrically neutral, and so attractive and repulsive Coulomb forces nearly cancel. Gravitational force on a large scale dominates interactions between large objects because it is always attractive, while Coulomb forces tend to cancel. Section Summary
Use Figure 3 as a reference in the following questions. Figure 3 shows a schematic representation of the outer electron cloud of a neutral water molecule. The electrons spend more time near the oxygen than the hydrogens, giving a permanent charge separation as shown. Water is thus a polar molecule. It is more easily affected by electrostatic forces than molecules with uniform charge distributions.
Problems & Exercises
Coulomb’s law: the mathematical equation calculating the electrostatic force vector between two charged particles Coulomb force: another term for the electrostatic force electrostatic force: the amount and direction of attraction or repulsion between two charged bodies
2. (a) 0.263 N; (b) If the charges are distributed over some area, there will be a concentration of charge along the side closest to the oppositely charged object. This effect will increase the net force. 4. The separation decreased by a factor of 5. 8. [latex]\begin{array}{lll}F&=&k\frac{\mid{q}_{1}{q}_{2}\mid}{{r}^{2}}=ma\Rightarrow{a}=\frac{k{q}^{2}}{mr^{2}}\\\text{ }&=&\frac{\left(9.00\times{10}^{9}\text{ N}\cdot\text{m}^{2}/{\text{C}}^{2}\right){\left(1.60\times {10}^{-19}\text{ m}\right)}^{2}}{\left(1.67\times {10}^{-27}\text{ kg}\right){\left(2.00\times {10}^{-9}\text{ m}\right)}^{2}}\\ & =& 3.45\times {10}^{16}\text{ m/s}^{2}\end{array}\\[/latex] 9. (a) 3.2; (b) If the distance increases by 3.2, then the force will decrease by a factor of 10 ; if the distance decreases by 3.2, then the force will increase by a factor of 10. Either way, the force changes by a factor of 10. 11. (a) 1.04 × 10−9 C; (b) This charge is approximately 1 nC, which is consistent with the magnitude of charge typical for static electricity 14. 1.02×10−11 16. (a) 0.859 m beyond negative charge on line connecting two charges; (b) 0.109 m from lesser charge on line connecting two charges |