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Unit 1: Electric Charge & Coulomb's Law

What You'll Learn

In this unit you will master the fundamental properties of electric charge — quantization, conservation, and the two-sign convention — then derive and apply Coulomb's law in scalar and vector form. You will practice superposition for multi-charge systems and distinguish conductors from insulators at the atomic level. Use the Desmos panel to plot force-vs-distance curves and verify the inverse-square relationship.
Definition

Electric Charge

Electric charge (qq) is an intrinsic property of matter that causes it to experience a force in an electromagnetic field. Charge is quantized: the smallest free charge is the elementary charge e=1.602×10−19  Ce = 1.602 \times 10^{-19}\;\text{C}. Any observable charge satisfies q=neq = ne where nn is an integer. Charge is a scalar — positive or negative — and obeys a conservation law: the net charge of an isolated system never changes.
Definition

Conductors & Insulators

A conductor is a material in which charge carriers (typically electrons in metals) move freely under an applied electric field. In electrostatic equilibrium, all excess charge resides on the surface and E⃗=0\vec{E} = 0 inside. An insulator (dielectric) has charges bound to atoms; they can polarize but not flow freely. Semiconductors fall between the two extremes, with conductivity that depends on temperature and doping.
Conductor−−−−−−−−−−E = 0 insideInsulator−−−−−−−−−Charges fixed in placeCharges free to moveCharges migrate to surface in conductors
In a conductor excess charges migrate to the surface so the interior field vanishes. In an insulator charges remain fixed where they are deposited.
Definition

Coulomb's Law (Scalar Form)

The magnitude of the electrostatic force between two point charges q1q_1 and q2q_2 separated by distance rr is F=ke∣q1q2∣r2F = k_e \frac{|q_1 q_2|}{r^2} where Coulomb's constant ke=14πε0≈8.99×109  N\cdotpm2/C2k_e = \frac{1}{4\pi\varepsilon_0} \approx 8.99 \times 10^9\;\text{N·m}^2/\text{C}^2 and ε0=8.854×10−12  C2/(N\cdotpm2)\varepsilon_0 = 8.854 \times 10^{-12}\;\text{C}^2/(\text{N·m}^2) is the permittivity of free space.
The force on charge q2q_2 due to charge q1q_1 is F⃗12=keq1q2r2 r^12\vec{F}_{12} = k_e \frac{q_1 q_2}{r^2}\,\hat{r}_{12} where r^12\hat{r}_{12} points from q1q_1 to q2q_2. The force is repulsive (F⃗\vec{F} along r^\hat{r}) for like signs and attractive (opposite r^\hat{r}) for unlike signs. Newton's third law guarantees F⃗21=−F⃗12\vec{F}_{21} = -\vec{F}_{12}.
Coulomb's force is an inverse-square law: doubling the distance reduces the force to one-quarter.
+q₁+q₂rF₁₂F₂₁|F| = kq₁q₂ / r²Like charges repel — equal magnitude, opposite direction
Two like charges repel with equal-magnitude, opposite-direction forces. The dashed line indicates the separation rr.
The net force on a charge q0q_0 due to NN other point charges is the vector sum of the individual Coulomb forces: F⃗net=∑i=1Nkeq0qir0i2 r^0i\vec{F}_{\text{net}} = \sum_{i=1}^{N} k_e \frac{q_0 q_i}{r_{0i}^2}\,\hat{r}_{0i} Each pair interacts independently; intermediate charges do not screen one another.
Always break forces into components ($x$, $y$) before summing; magnitudes alone are not enough.
Definition

Methods of Charging

Objects can be charged by friction (triboelectric effect), conduction (direct contact transfers charge), or induction (a nearby charged object polarizes a conductor; grounding then removes one sign of charge). In induction the inducing object never touches the target, so its own charge is unchanged.
Definition

Polarization of Insulators

Even though charges in an insulator cannot flow, an external electric field slightly shifts the electron clouds relative to nuclei, creating tiny induced dipoles. The net effect is a layer of bound surface charge that partially cancels the applied field inside the material. This phenomenon is called dielectric polarization and is characterized by the polarization vector P⃗\vec{P}.
Ex

Example — Force Between Two Point Charges

Two charges q1=+3  μCq_1 = +3\;\mu\text{C} and q2=−5  μCq_2 = -5\;\mu\text{C} are 0.20  m0.20\;\text{m} apart. Find the magnitude and direction of the force on q2q_2.
Solution Steps
Ex

Example — Superposition with Three Collinear Charges

Charges qA=+2  μCq_A = +2\;\mu\text{C}, qB=−4  μCq_B = -4\;\mu\text{C}, and qC=+1  μCq_C = +1\;\mu\text{C} sit on the xx-axis at x=0x = 0, x=0.30  mx = 0.30\;\text{m}, and x=0.50  mx = 0.50\;\text{m}. Find the net force on qBq_B.
Solution Steps
Ex

Example — 2-D Superposition: Equilateral Triangle

Three identical charges q=+4  μCq = +4\;\mu\text{C} sit at the vertices of an equilateral triangle with side a=0.10  ma = 0.10\;\text{m}. Find the net force on the charge at the top vertex.
Solution Steps

Desmos Exploration — Inverse-Square Law

Open the Desmos panel and plot F(r)=k⋅q1⋅q2/r2F(r) = k \cdot q_1 \cdot q_2 / r^2. Set sliders for q1q_1, q2q_2, and k=8.99×109k = 8.99 \times 10^9. Observe how the force curve steepens as charges increase and flattens as rr grows.
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Practice Problem
medium
Two protons (q=1.6×10−19  Cq = 1.6 \times 10^{-19}\;\text{C}) are separated by 1.0×10−15  m1.0 \times 10^{-15}\;\text{m}. What is the electrostatic force between them?
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Practice Problem
hard
Charge A=+6  μCA = +6\;\mu\text{C} is at the origin. Charge B=−3  μCB = -3\;\mu\text{C} is at (0.40,0)(0.40,0). Where on the xx-axis is the net force on a test charge zero?
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Practice Problem
medium
A rubber rod acquires charge −4.8×10−7  C-4.8 \times 10^{-7}\;\text{C}. How many excess electrons does it have?
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Practice Problem
medium
A positively charged rod is brought near (but not touching) a grounded metal sphere. The ground wire is removed, then the rod is removed. The sphere's final charge is:
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Practice Problem
medium
Two identical conducting spheres carry +5Q+5Q and −3Q-3Q. After touching and separating, each has charge:
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Practice Problem
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Fixed charges +Q+Q and +4Q+4Q are separated by dd. A third charge is in equilibrium at distance from +Q+Q of:
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Practice Problem
medium
If the distance between two charges is tripled, the electrostatic force:
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Practice Problem
hard
q1=+4  μCq_1 = +4\;\mu\text{C} at origin, q2=−2  μCq_2 = -2\;\mu\text{C} at (3,0)  m(3,0)\;\text{m}, q3=+1  μCq_3 = +1\;\mu\text{C} at (0,4)  m(0,4)\;\text{m}. Magnitude of net force on q1q_1?
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