Electric Field and Coulomb's Law

⚡ Electric Charge

Part 1 of 7 — Electric Charge

• Charge is quantized: $q = ne$ where $e = 1.6 \times 10^{-19}$ C
• Charge is conserved in all processes
• Like charges repel, unlike charges attract
• Conductors allow charge flow; insulators do not

Worked Example

How many excess electrons on an object with charge $-3.2 \times 10^{-19}$ C?

$n = \frac{q}{e} = \frac{3.2 \times 10^{-19}}{1.6 \times 10^{-19}} = 2$ electrons ✅

Concept Check 🎯

Electric Charge 🧮

1. $q = -3.2 \times 10^{-19}$ C. How many excess electrons?

2. An object has $q = -8.0 \times 10^{-19}$ C. How many excess electrons?

3. Two spheres: $+5$ μC and $+1$ μC. After touching, each has ____ μC.

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⚡ Coulomb's Law

Part 2 of 7 — Coulomb's Law

$F = k\frac{|q_1||q_2|}{r^2}$

where $k = 8.99 \times 10^9$ $N\cdot m^{2}/C^{2}$ $= \frac{1}{4\pi\epsilon_0}$

• Force is along the line connecting the charges
• Attractive for opposite charges, repulsive for like charges

Worked Example

Two $+2$ μC charges are 0.3 m apart. Find the force.

$F = k\frac{q_1 q_2}{r^2} = (9 \times 10^9)\frac{(2 \times 10^{-6})^2}{(0.3)^2}$

$= (9 \times 10^9)\frac{4 \times 10^{-12}}{0.09} = 0.4$ N (repulsive) ✅

Concept Check 🎯

Coulomb's Law 🧮

1. If the distance between two charges is halved, the force increases by a factor of ___

2. Two $+1$ μC charges are 0.1 m apart. $F = k(10^{-12})/(0.01) = 9 \times 10^9 \times 10^{-10}$ = ____ N. (Give the answer as a decimal.)

3. The Coulomb constant $k$ is approximately ____ $\times 10^9$ $N\cdot m^{2}/C^{2}$.

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⚡ Superposition Principle

Part 3 of 7 — Superposition Principle

The net force on a charge is the vector sum of all individual Coulomb forces:

$\vec{F}_{net} = sum_i \vec{F}_i = sum_i k\frac{q q_i}{r_i^2}\hat{r}_i$

Each force is calculated independently, then added as vectors.

Worked Example

Three charges on a line: $+q$ at $x = 0$, $+q$ at $x = d$, $-q$ at $x = 2d$. Force on the middle charge?

From left (+q): $F_1 = kq^2/d^2$ (rightward, repulsive)

From right (-q): $F_2 = kq^2/d^2$ (rightward, attractive)

$F_{net} = 2kq^2/d^2$ (rightward) ✅

Concept Check 🎯

Superposition Principle 🧮

1. Equal charges $+q$ are placed at $x = -d$ and $x = +d$. Net force on a charge at the origin? (Give as a multiple of $kq^2/d^2$.)

2. A charge at the origin feels $F_1 = +3$ N and $F_2 = -1$ N along the x-axis. Net force (N)?

3. Three charges in a row: $+q$ at 0, $+q$ at $d$. Force from left on right = $kq^2/d^2$. If a third $+q$ is at $2d$, force from it on the middle charge = $kq^2/d^2$. But wait, that force is attractive or repulsive? It's repulsive... Hmm. Simply: how many charges are present?

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⚡ Electric Field

Part 4 of 7 — Electric Field

$\vec{E} = \frac{\vec{F}}{q_0} = k\frac{q}{r^2}\hat{r}$

• $E$ points away from positive charges, toward negative charges
• SI unit: N/C = V/m
• The electric field is a vector field: it has magnitude and direction at every point

Worked Example

Find $E$ at 0.5 m from a $+4$ μC charge.

$E = kq/r^2 = (9 \times 10^9)(4 \times 10^{-6}) / (0.25) = 144{,}000$ N/C ✅

Concept Check 🎯

Electric Field 🧮

1. If the distance from a charge triples, $E$ decreases by a factor of ___.

2. A charge $q$ in a field $E = 10$ N/C feels $F = 50$ N. What is $q$ (in C)?

3. A $+2$ C charge in a field $E = 10$ N/C. What is the force (N)?

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⚡ Electric Field Lines

Part 5 of 7 — Field Lines

Rules for electric field lines:

1. Start on positive charges, end on negative charges
2. Never cross each other
3. Density indicates field strength
4. Perpendicular to conducting surfaces
5. Number of lines proportional to charge magnitude

Worked Example

A $+2q$ charge and a $-q$ charge are nearby. Describe the field lines.

• Twice as many lines leave $+2q$ as enter $-q$
• Some lines from $+2q$ extend to infinity
• Near each charge, lines are radial
• Lines curve from $+2q$ toward $-q$ ✅

Concept Check 🎯

Field Lines 🧮

1. A charge of $+2q$ has twice as many field lines as a charge of $+q$. If $+q$ has 4 lines, how many does $+2q$ have? Give answer divided by 4.

2. At a point where field lines are absent, the field strength is ___ N/C.

3. A $+2q$ charge has 8 field lines. A $-q$ charge has 4 field lines entering. How many lines escape to infinity?

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⚡ Problem-Solving Workshop

Part 6 of 7 — Problem-Solving Workshop

Strategy for Electrostatics

1. Draw a diagram with all charges and distances
2. For forces: apply Coulomb's law to each pair
3. For fields: find $\vec{E}$ from each charge, then superpose
4. Use symmetry to simplify
5. Set up coordinate system and resolve vectors

Worked Example

Two charges $+3$ μC and $-3$ μC are 0.4 m apart. Find the field at the midpoint.

Both fields point from $+$ to $-$ (same direction at midpoint).

$E_{each} = k(3 \times 10^{-6})/(0.2)^2 = 675{,}000$ N/C

$E_{total} = 2 \times 675{,}000 = 1{,}350{,}000$ N/C (toward $-$ charge) ✅

Concept Check 🎯

Problem-Solving Workshop 🧮

1. Two identical charges $+q$ are equidistant from a point P, on opposite sides. Net field at P? (units of $kq/r^2$)

2. $F = kq_1q_2/r^2$. If $k = 9 \times 10^9$, $q_1 = q_2 = 2 \times 10^{-6}$ C, $r = 1$ m. $F$ (N)? (Give as integer in mN, i.e., $\times 10^{-3}$)

3. If the distance between charges is halved, force increases by a factor of ___

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⚡ Review & Applications

Part 7 of 7 — Review & Applications

Key Formulas

• $F = kq_1q_2/r^2$ (Coulomb's Law)
• $\vec{E} = k q/r^2 \hat{r}$ (Electric field)
• $\vec{F} = q\vec{E}$ (Force on charge in field)
• Superposition: $\vec{F}_{net} = \sum \vec{F}_i$
• $e = 1.6 \times 10^{-19}$ C, $k = 9 \times 10^9$ $N\cdot m^{2}/C^{2}$

Worked Example

An electron is in a uniform field $E = 100$ N/C. Find its acceleration.

$F = eE = (1.6 \times 10^{-19})(100) = 1.6 \times 10^{-17}$ N

$a = F/m = 1.6 \times 10^{-17} / 9.1 \times 10^{-31} \approx 1.76 \times 10^{13}$ $m/s^{2}$ ✅

Concept Check 🎯

Review & Applications 🧮

1. Doubling both charges in Coulomb's law increases the force by a factor of ___

2. A charge of 2 C in a field of 5 N/C. Force (N)?

3. $k \approx$ ___ $\times 10^9$ $N\cdot m^{2}/C^{2}$

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Practice

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