Electric Charge and Coulomb's Law: Quick Revision

A brief note on Electric Charge and Coulomb's Law. Read it before a test. Try each question at the end before you open its answer.

Charge

• Two kinds: positive and negative. Like charges repel. Unlike charges attract.
• Rubbing: only electrons move. The body that gains electrons is negative.
• Unit: the coulomb, $1\ \text{C} = 1\ \text{A s}$. $e = 1.6 \times 10^{-19}$ C.
• Conductors and insulators: free electrons move in a conductor. Charge stays where it was put on an insulator.
• Contact: the body gets the same sign as the rod. Induction: the body gets the opposite sign, and the rod keeps its charge.
• Additive: add charges with their signs.
• Conserved: the total charge of an isolated system does not change.
• Quantised: $Q = ne$, with $n$ a whole number.
• Identical metal spheres that touch share the total charge equally.

Coulomb's law

• Force between point charges: $F = k\dfrac{q_1q_2}{r^2}$, along the line that joins them.
• The constant: $k = \dfrac{1}{4\pi\varepsilon_0} = 9.0 \times 10^9\ \text{N m}^2/\text{C}^2$, with $\varepsilon_0 = 8.85 \times 10^{-12}\ \text{C}^2/(\text{N m}^2)$.
• Direction: the two forces are equal and opposite. Like charges repel and unlike charges attract.
• Distance: double $r$ and the force falls to a quarter.
• In a medium: $F = \dfrac{1}{K}\left(k\dfrac{q_1q_2}{r^2}\right)$. For water $K \approx 80$.
• Compared with gravity: between an electron and a proton the electric force is about $10^{39}$ times larger.

Several charges

• Superposition: other charges do not change the force between two charges. $\vec F = \vec F_1 + \vec F_2 + \dots$
• On a line: choose a positive direction and add the forces with signs.
• In a plane: add components, then $F = \sqrt{F_x^2 + F_y^2}$.
• Zero force between two like charges: $\dfrac{x}{d - x} = \sqrt{\dfrac{q_1}{q_2}}$, nearer the smaller charge.

A worked example

A charge of $+8\ \mu\text{C}$ and a charge of $-2\ \mu\text{C}$ are $0.40$ m apart in air. Find the force between them.

1. Sizes in SI units. $q_1 = 8 \times 10^{-6}$ C and $q_2 = 2 \times 10^{-6}$ C.
2. The size. $F = \dfrac{kq_1q_2}{r^2}$. The top is $9.0 \times 10^9 \times 8 \times 10^{-6} \times 2 \times 10^{-6}\ \text{N m}^2 = 0.144\ \text{N m}^2$. The bottom is $(0.40\ \text{m})^2 = 0.16\ \text{m}^2$. So $F = 0.9\ \text{N}$.
3. The direction. The charges are unlike, so each pulls the other with $0.9$ N.

Practise with new numbers

Check yourself

1. Two charges repel with a force of $20$ N. They are put in a liquid with a dielectric constant of $4$, at the same distance. What is the force now?

2. How many electrons make a charge of $5.0$ nC?

3. A charge $+q$ is placed midway between a charge $+Q$ and a charge $-Q$. Which way does the net force on $+q$ point?