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Electric Field and Coulomb's Law

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Electric Field and Coulomb's Law - Complete Interactive Lesson

Part 1: Electric Charge

⚡ 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 🧮

$q = -3.2 \times 10^{-19}$ C. How many excess electrons?
An object has $q = -8.0 \times 10^{-19}$ C. How many excess electrons?

Concept Check 🔍

Practice

#	Concept	Key Fact
1	Elementary charge	$e = 1.6 \times 10^{-19}$ C
2	Charge conservation	Total charge is constant
3	Quantization	Charge comes in integer multiples of

Challenge Question 📋

Part 2: Coulomb's Law

⚡ Coulomb's Law

Part 2 of 7 — Coulomb's Law

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

Part 3: Superposition Principle

⚡ Superposition Principle

Part 3 of 7 — Superposition Principle

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

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

Part 4: Electric Field

⚡ Electric Field

Part 4 of 7 — Electric Field

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

Part 5: Field Lines

⚡ Electric Field Lines

Part 5 of 7 — Field Lines

Rules for electric field lines:

Start on positive charges, end on negative charges
Never cross each other
Density indicates field strength
Perpendicular to conducting surfaces
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

Part 6: Problem-Solving Workshop

⚡ Problem-Solving Workshop

Part 6 of 7 — Problem-Solving Workshop

Strategy for Electrostatics

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

Worked Example

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

Part 7: Review & Applications

⚡ Review & Applications

Part 7 of 7 — Review & Applications

Key Formulas

$F = kq_1q_2/r^2$ (Coulomb's Law)

1.6×10−193.2×10−19=

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

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 🧮

If the distance between two charges is halved, the force increases by a factor of ___
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.)
The Coulomb constant $k$ is approximately ____ $\times 10^9$ $N\cdot m^{2}/C^{2}$ .

Concept Check 🔍

Practice

#	Charges	Distance
1	$+1$ μC, $-1$ μC	0.1 m
2	$+3$ μC, $+3$ μC	0.3 m
3	$+2$ μC, $-4$ μC	0.2 m

Challenge Question 📋

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 🧮

Equal charges $+q$ are placed at $x = -d$ and $x = +d$ . Net force on a charge at the origin? (Give as a multiple of $kq^2/d^2$ .)
A charge at the origin feels $F_1 = +3$ N and $F_2 = -1$ N along the x-axis. Net force (N)?
Three charges in a row: $+q$ at 0, $+q$ at $d$ . Force from left on right = $kq^2/d^2$ . If a third is at , force from it on the middle charge = . But wait, that force is attractive or repulsive? It's repulsive... Hmm. Simply: how many charges are present?

Concept Check 🔍

Practice

#	Configuration	Method
1	Three collinear charges	Add forces with signs
2	Triangle of charges	Vector components
3	Two charges + test charge	Superposition

Challenge Question 📋

$E$ 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 🧮

If the distance from a charge triples, $E$ decreases by a factor of ___.
A charge $q$ in a field $E = 10$ N/C feels $F = 50$ N. What is $q$ (in C)?
A $+2$ C charge in a field $E = 10$ N/C. What is the force (N)?

Concept Check 🔍

Practice

#	Concept	Formula
1	Point charge field	$E = kq/r^2$
2	Force on a charge	$F = qE$
3	Superposition of fields	$\vec{E}_{net} = \sum \vec{E}_i$

Challenge Question 📋

Some lines from

+2q

extend to infinity

Near each charge, lines are radial

Lines curve from

+2q

toward

-q

✅

Concept Check 🎯

Field Lines 🧮

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.
At a point where field lines are absent, the field strength is ___ N/C.
A $+2q$ charge has 8 field lines. A $-q$ charge has 4 field lines entering. How many lines escape to infinity?

Concept Check 🔍

Practice

#	Rule	Application
1	Lines start/end on charges	Identify sources/sinks
2	Density = field strength	Closer lines = stronger field
3	Lines don't cross	Unique field direction at each point

Challenge Question 📋

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 🧮

Two identical charges $+q$ are equidistant from a point P, on opposite sides. Net field at P? (units of $kq/r^2$ )
$F = kq_1q_2/r^2$ . 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., $\times 10^{-3}$ )
If the distance between charges is halved, force increases by a factor of ___

Concept Check 🔍

Practice

#	Problem Type	Strategy
1	Midpoint field (dipole)	Fields add (both point same way)
2	Equilibrium position	Set $F_1 = F_2$ , solve for $r$
3	Force between three charges	Superposition

Challenge Question 📋

\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}

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

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

Concept Check 🎯

Review & Applications 🧮

Doubling both charges in Coulomb's law increases the force by a factor of ___
A charge of 2 C in a field of 5 N/C. Force (N)?
$k \approx$ ___ $\times 10^9$ $N\cdot m^{2}/C^{2}$

Concept Check 🔍

Practice

#	Topic	Key Fact
1	Coulomb's law	Inverse-square law
2	Electric field	$E = F/q$
3	Superposition	Vector sums

Challenge Question 📋