Coulomb's Law
The metric unit of charge is the Coulomb (C).
The symbol for charge is Q or q.
The charge of a single electron is -1.60 x 10-19 C.
The charge of a single proton is +1.60 x 10-19 C.
A Coulomb is equal to the charge of 6.25 x 10 18 electrons or protons.
It’s important to remember the Greek prefixes.
Because the Coulomb is a large amount of charge, one often uses micro-Coulombs or milli-Coulombs.
1 mC = 10-3 C
1 µC = 10-6 C
Exercise: What is the charge of 1.35 x 1017 electrons?
1.35•1017e • 1.6•10-19C/e = 2.06•10-2C or 20.6µC
Charged objects exert forces on one another according to Coulomb’s Law:
F = k (q1 q2) / d2
F : force exerted between the two charges.
q1 and q2 : two charges.
d : distance between the two charges
k : electrical constant, 9•109 N m2 / C2
The force between two charged objects can be either attractive or repulsive,
depending on whether the charges are like or unlike.
Exercise: A 0.02C and 0.03C charge are 5m apart. What is the force between them?
F = k (q1 q2) / d2
F = 9•10 N m2 / C2 (0.02C • 0.03C) / (5m)2
F = 2.2•105N and repulsive
9
Exercise: A force of 1.6 x 10-3 N exists between 2 charges; 1.3 µC and 3.5µC.
How far apart are they?
F = k (q1 q2) / d2
d = √(k q1 q2 / F)
d = √((9•109 N m2 / C2) (1.3•10-6C) (3.5•10-6C) / (1.6•10-3N))
d = 5.1m
Electric Force and Gravity: Both gravity and the electric force are fundamental forces. The equations for
the gravity and the electromagnetic force have the same form; they are both inverse square relationships.
Fe = k (q1 q2) / d2
Fg = G (m1 m2) / d2
There are four main differences between them:
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•
•
•
Gravity is always attractive; the electromagnetic force can be either attractive or repulsive.
Gravity is much weaker.
Gravity has a much greater range within which it is a significant force.
The electric force can be shielded, blocked, or cancelled; gravity cannot.
Gravity extends out to great distances. The sun is 92 million miles away from Earth, yet the force of gravity is
large enough to lock Earth in its orbit. The force between two charges drops off very quickly with distance.
This is because the magnitude of the two charges is very small – at the most, maybe a few Coulombs. But with
gravity, we are dealing with enormous masses - the mass of the sun is 2•1030 kg! Because of the large masses
involved, the force of gravity reaches out over distances of billions and even trillions of kilometers.
Exercise: Compare gravity and Coulomb attraction within a hydrogen atom,
which consists of a proton and an electron.
qp = 1.60•10-19 C
Fg = 6.67•10
-11
qe = -1.60•10-19 C
separation distance = 5.3•10-11 m
me = 9.1•10 –31 kg
mp = 1.7 x 10 –27 kg.
Fg = G (m1 m2) / d2
N m / kg (9.1•10 –31 kg • 1.7 x 10 –27 kg) / (5.3•10-11 m)2
Fg = 3.7•10-47N
2
2
Fe = k (q1 q2) / d2
Fe = 9•109 N m2 / C2 (1.60•10-19 C • -1.60•10-19 C) / (5.3•10-11 m)2
Fe = 8.2•10-8N
Gravity is much weaker; the electromagnetic force is 2.2 x 1039 times bigger!
Superposition Principle: When three or more charges interact, add the forces as vectors.
Exercise: Find the force on the charge at the origin; the graph is in cm.
F1 2 = k Q1 Q2 / d1 22
F1 2 = (9•109) (-2•10-9) (5•10-9) / (0.04)2
F1 2 = -5.63•10-5N
F1 3 = k Q1 Q3 / d1 32
F1 3 = (9•109) (-2•10-9) (6•10-9) / (0.03)2
F1 3 = -1.2•10-4N
∑F = √(( F1 2)2 + (F1 3)2)
∑F = √(( -5.63•10-5N)2 + (-1.2•10-4N)2)
∑F = 1.32•10-4N
ø = tan-1 ( 1.2•10-4N / 5.63•10-5N) = 65˚
Exercise:
Two 25 g metal spheres have identical positive charges.
They down from light strings that are 32 cm long.
Each string forms a 7˚ angle with the vertical.
What is the magnitude of the charges?
First draw a FBD for the forces on one of the spheres.
Tension holds it up, electric force pushes it sideways,
and weight pulls down.
The sum of these forces must be zero.
In the X direction : Fe = T sin θ
In the Y direction : T cos θ = mg
T = mg / cos θ
T = 0.025kg • 9.8 m/s2 / cos 7˚ = 0.247N
Fe = T sin θ
Fe = 0.247N sin 7˚ = 0.0301N
The distance between the two charges is 2 (0.32m) sin 7˚ = 7.8cm
Fe = k (qq) / d2
Remember that the charges are the same.
q = √(Fe d2 / k)
q = √(0.0301 (7.8)2 / 9•109)
q = 1.43•10-7C