Physics · Ch 1 — Electrostatics
COULOMB'S LAW
COULOMB'S LAW
In 1785, Charles-Augustin de Coulomb deduced the expression for the force between two stationary point charges q1 and q2 separated by a distance r in vacuum. The force on charge q2 due to charge q1 is F21 = k (q1 q2 / r^2) r-hat-12, where r-hat-12 is the unit vector pointing from q1 to q2, and k is a proportionality constant. In SI units, k = 1/(4 pi epsilon0), with epsilon0 the permittivity of free space, epsilon0 = 8.85x10^-12 C^2 N^-1 m^-2, giving k = 9x10^9 N m^2 C^-2. Important aspects of Coulomb's law: (i) the force is directly proportional to the product of the magnitudes of the two charges and inversely proportional to the square of the distance between them; (ii) the force lies along the line joining the two charges; the force on q1 due to q2, F12, equals -F21, consistent with Newton's third law; (iii) one coulomb of charge is an enormous amount -- two 1 C charges 1 m apart in vacuum repel with a force of 9x10^9 N, roughly the weight of a million tonnes, which is why everyday charges are measured in microcoulombs or nanocoulombs; (iv) in a medium of permittivity epsilon (epsilon = epsilon_r epsilon0, with epsilon_r the relative permittivity, equal to 1 for vacuum/air and greater than 1 for every other medium), the force is F = q1 q2/(4 pi epsilon r^2), always weaker than the vacuum-value since epsilon > epsilon0; (v) Coulomb's law has the same inverse-square structure as Newton's law of gravitation, but differs from it in three important ways -- gravity is always attractive while the Coulomb force can be attractive or repulsive; the Coulomb constant k = 9x10^9 N m^2 C^-2 is vastly larger than the gravitational constant G = 6.67x10^-11 N m^2 kg^-2, so electrostatic forces dominate gravitational ones for small charged objects; and gravity is independent of the surrounding mediu …
What this figure shows. Two positive point charges q1 and q2 sit a distance r apart on a line, with the unit vector r-hat-12 drawn from q1 toward q2 along that line. The force F21 that charge q1 exerts on charge q2 is drawn pointing away from q1, along the same line as r-hat-12, correctly showing that like charges repel and that the Coulomb force always acts along the straight line joining the two point charges, never …
Worked out. Two point charges 1 m apart are considered in three cases. (a) With q1 = +2 uC and q2 = +3 uC, both positive, the force is repulsive; using F = k q1 q2 / r^2 with k = 9x10^9, the magnitude works out to 5.4x10^-3 N, pushing the charges apart along the line joining them, and by Newton's third law the force on q1 due to q2 is equal and opposite to the force on q2 due to q1. (b) With q1 = +2 uC and q2 = -3 uC, unlike charges, the force has the same magnitude 5.4x10^-3 N but is now attractive, pulling the charges together. (c) With the same unlike charges now immersed in water (relative permittivity er = 80), the force is reduced by a factor of er to 0.675x10^-4 N -- water's very high relative permittivity is exactly why it is such an effective solvent for ionic salts like NaCl, since it drastically weake …
Worked out. Two identical small charged spheres, each of mass 1 g, hang from two strings of length 10 cm from a common point, coming to rest with each string making 30 degrees with the vertical, and g = 10 m/s^2. Resolving each sphere's weight, string tension, and the mutual Coulomb repulsion into horizontal and vertical components and applying Newton's second law (net force zero in equilibrium) gives tan(theta) = Fe/(mg), which combined with r = 2Lsin(theta) for the separation between the spheres yields Fe = k q^2/(2Lsin theta)^2. Solving this for q using the given numbers produces q = 8.01x10^-8 C = 80.1 nC on each sphere -- the mutual repulsion is exactly strong enough to hold the two strings …
Worked out. For a proton and an electron in a hydrogen atom, separated by 5.3x10^-11 m, both carrying charge magnitude 1.6x10^-19 C, the attractive electrostatic force works out to Fe = k e^2/r^2 = 8.2x10^-8 N, while the attractive gravitational force between their masses (electron 9.1x10^-31 kg, proton 1.6x10^-27 kg) works out to FG = G me mp/r^2 = 3.4x10^-47 N. The ratio Fe/FG is about 2.4x10^39, showing that the electrostatic force binding the atom together is enormously (about 10^39 times) stronger than the gravitational attraction between the same two particles -- which is exactly why a charged comb can lift a small piece of paper against the pull of the entire Earth's gravity, and why gravity can safely be ignored compared to electro …
What this figure shows. A charged comb is shown lifting small scraps of paper off a table, with the upward electrostatic attractive force Fe on the paper drawn much longer than the downward gravitational force FG that the whole Earth exerts on the same paper. The picture is the everyday, human-scale illustration of the enormous Fe/FG ratio computed in Example 1.4: even though the entire mass of the Earth is pulling the paper down, the comb's comparatively tiny induced charge produces an electrostatic force strong enough to overcome it, because the Coulomb force between charges at short range vastly exceeds the gravitat …