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Electric Charge and Coulomb's Law
Electromagnetism
Electric charge is one of the basic properties of matter, carried by protons and electrons. Charges exert forces on each other across empty space, and Coulomb's law gives that force a precise size and direction.
What electric charge is
Electric charge comes in two kinds, positive and negative. Protons carry a positive charge and electrons carry a negative charge, and the size of one elementary charge is about 1.60 x 10^-19 coulombs. Charge is measured in coulombs, symbol C, in the SI system. An object is neutral when it has equal amounts of positive and negative charge, positively charged when it has lost electrons, and negatively charged when it has gained extra electrons. Charge is conserved: it can move from one object to another, but the total charge in an isolated system never changes. Rubbing a balloon on hair transfers electrons to the balloon, which is why it can then stick to a wall.
Coulomb's law
Coulomb's law states that the electric force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. The force acts along the line joining the charges: like charges repel and opposite charges attract. Because of the inverse-square dependence, doubling the distance makes the force four times smaller, while doubling one charge doubles the force. The constant k is 8.99 x 10^9 N m^2/C^2 in a vacuum, and it is large because the coulomb is a very large unit of charge. The law works best for point charges or small charged spheres whose separation is large compared with their size.
Where you see it
Static electricity is the most familiar example: clothes cling in a dryer, hair stands up after combing, and a spark jumps when you touch a doorknob after walking on carpet. Lightning is a dramatic discharge between charged regions of a cloud or between a cloud and the ground. On a smaller scale, electric forces hold atoms and molecules together, drive currents in wires, and operate inkjet printers and photocopiers, which steer tiny charged droplets or toner particles with electric fields. Engineers must also control unwanted charge build-up, because a sudden discharge can damage sensitive electronic components on a factory floor.
📝 Worked example: force between two small charges
Two small charges are in air. q1 = 2.0 x 10^-6 C and q2 = 3.0 x 10^-6 C, separated by r = 0.10 m. Use F = k q1 q2 / r^2. Substitute: F = (8.99 x 10^9) x (2.0 x 10^-6) x (3.0 x 10^-6) / (0.10)^2. Numerator: (8.99 x 10^9) x (6.0 x 10^-12) = 5.39 x 10^-2. Denominator: (0.10)^2 = 0.010. F = 5.39 x 10^-2 / 0.010 = 5.39 N. The force is repulsive, because both charges are positive, so each charge feels a 5.39 N push away from the other.
Common mistakes
- Confusing charge with current: charge is a property measured in coulombs, while current is the rate of flow of charge measured in amperes.
- Forgetting the square on distance: the force falls as 1/r^2, not 1/r, so doubling the distance quarters the force.
- Dropping the sign: the sign of the charges tells you attraction or repulsion, but the magnitude formula uses the sizes of the charges.
Practice questions
- Two charges of +4.0 microC and +2.0 microC are 0.20 m apart. What is the magnitude of the force between them?
Show answer
F = (8.99 x 10^9)(4.0 x 10^-6)(2.0 x 10^-6)/(0.20)^2 = 7.19 x 10^-2 / 0.040 = 1.80 N, repulsive.
- If the distance between two charges is tripled, by what factor does the force change?
Show answer
The force falls by 3^2 = 9, so it becomes one-ninth of its original value.
- How many electrons make up a charge of -1.60 x 10^-18 C?
Show answer
n = q/e = 1.60 x 10^-18 / 1.60 x 10^-19 = 10 electrons.
Frequently asked questions
Can charge be created or destroyed?
No. Charge is conserved; it can only be transferred, and positive and negative charges are always created or destroyed in equal pairs.
Why is the coulomb such a large unit?
Because one electron carries only 1.60 x 10^-19 C, so about 6.24 x 10^18 electrons are needed to make one coulomb.