Electric field — definition, field lines, superposition

1. Definition

WHY the limit $q_0\to 0$ ? A real test charge would push on the source charges and move them, changing the very field we want to measure. We imagine the test charge vanishingly small so it probes without disturbing.

Deriving the field of a point charge — from scratch

WHAT we know (Coulomb's law): the force on charge $q_0$ at distance $r$ from a point charge $q$ is $\vec F = \frac{1}{4\pi\varepsilon_0}\frac{q\,q_0}{r^2}\,\hat r .$

HOW we get the field: divide out the test charge per the definition. $\vec E = \frac{\vec F}{q_0} = \frac{1}{4\pi\varepsilon_0}\frac{q\,q_0}{r^2 q_0}\hat r$

The force on any charge $Q$ placed at that point is then simply $\vec F = Q\vec E$ . (Positive $Q$ feels force along $\vec E$ ; negative $Q$ , opposite.)

2. Field lines (Dual coding)

Figure — Electric field — definition, field lines, superposition

WHY does density mean strength? For a point charge, all lines spread out over a sphere of area $4\pi r^2$ . As $r$ grows the same number of lines spread over a bigger sphere, so density $\propto 1/r^2$ — exactly how $E$ falls off. The geometry of lines automatically encodes the inverse-square law.

3. Superposition

WHY is this allowed? Coulomb's law is linear in charge: double a source, double its force, and forces themselves add as vectors (Newton). Linearity → fields add. (This fails only in extreme regimes outside classical EM.)

4. Worked examples

Worked example (b) Two equal charges

$+q$ on the x-axis at $\pm a$ — field on the y-axis at height $y$ Step 1 — magnitude from each charge. Distance $r=\sqrt{a^2+y^2}$ , so $E_1=\dfrac{kq}{a^2+y^2}$ . Why this $r$ ? Pythagoras from charge to the point.

Step 2 — components. By symmetry the two horizontal $E_x$ cancel; vertical parts add. The vertical fraction is $\cos\theta = \dfrac{y}{\sqrt{a^2+y^2}}$ . Why cancel? Mirror symmetry about the y-axis flips $E_x$ but keeps $E_y$ .

Step 3 — total. $E_y = 2\,E_1\cos\theta = \frac{2kq}{a^2+y^2}\cdot\frac{y}{\sqrt{a^2+y^2}} = \frac{2kqy}{(a^2+y^2)^{3/2}}$ Why the $3/2$ power? One power of $r^2$ from Coulomb, half a power from the $\cos\theta$ projection.

Check the limit $y\gg a$ : $E_y\to \dfrac{2kq}{y^2}$ — looks like a single charge $2q$ far away. ✓ (Forecast-then-verify: we predicted "from far it's just total charge," and it is.)

5. Common mistakes

Recall Feynman: explain to a 12-year-old

Imagine every charged ball is like a tiny heater that warms the room around it — you feel "warmth" (force) even without touching the heater. The electric field is a map of how strong the warmth is and which way it pushes, at every spot in the room. If you put two heaters in the room, the warmth at any spot is just both warmths added together (as arrows). The arrows-on-paper are the field lines: crowded arrows = strong push.

Flashcards

What is the definition of electric field

\vec E

The force per unit positive test charge,

\vec E=\lim_{q_0\to0}\vec F/q_0

, units N/C.

Why take the limit

q_0\to0

in the definition?

So the test charge doesn't disturb (move) the source charges and change the field being measured.

Field of a point charge?

\vec E=\frac{1}{4\pi\varepsilon_0}\frac{q}{r^2}\hat r

, with

\hat r

from source to field point.

Why can't field lines cross?

At a crossing

\vec E

would have two directions, but the field has exactly one value at each point.

What does field-line density represent?

The magnitude (strength) of

\vec E

; denser = stronger.

State the superposition principle.

\vec E_{net}=\sum_i \vec E_i

— fields add as vectors, independent of other charges.

Why is superposition valid?

Coulomb's law is linear in charge and forces add as vectors.

On-axis field of two equal

+q

\pm a

, at height

y

E_y=\frac{2kqy}{(a^2+y^2)^{3/2}}

, horizontal parts cancel.

Why does a dipole's far field fall as

1/r^3

The

+

and

-

nearly cancel; only their small separation survives, steepening the falloff.

Direction of

\hat r

in the point-charge field formula?

From the source charge to the field point.

Connections

Coulomb's Law — the force law $\vec E$ is built from.
Gauss's Law — formalizes "line count" via flux $\Phi=\oint\vec E\cdot d\vec A$ .
Electric Potential — $\vec E=-\nabla V$ ; field lines are perpendicular to equipotentials.
Electric Dipole — example (c) is its starting point.
Continuous Charge Distributions — superposition becomes an integral.
Superposition Principle — same linearity idea in waves and circuits.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, electric field ka idea simple hai: koi bhi charge apne aas-paas ke space ko "charge" kar deta hai — matlab har point pe ek arrow ban jaata hai jo batata hai ki agar yahan ek chhota positive test charge rakha jaye to usko kitni aur kis direction me force milega. Field $\vec E = \vec F / q_0$ , units N/C. Limit $q_0 \to 0$ isliye lete hain taaki test charge khud source charges ko hila na de aur reading galat na ho.

Point charge ka field $E = kq/r^2$ — yeh seedha Coulomb's law se aata hai, bas test charge ko divide kar do, $q_0$ cancel ho jaata hai. Matlab field sirf source pe depend karta hai, na ki tum vahan kya rakhte ho. Field lines ek picture hain: arrow ki direction = field ki direction, aur lines ki density (kitni paas-paas hain) = field ki strength. Lines positive se nikalti hain, negative pe khatam hoti hain, aur kabhi cross nahi karti (warna ek point pe do directions ho jaayengi — impossible).

Superposition sabse powerful tool hai: agar bahut saare charges hain, to har ek ka field alag-alag nikaalo (jaise baaki hain hi nahi) aur sabko vector add kar do. Yaad rakho — magnitudes ko seedha mat jodo, pehle $x$ aur $y$ components nikaalo. Jaise do equal $+q$ ka y-axis pe field me horizontal parts cancel ho jaate hain, sirf vertical bachta hai: $E_y = 2kqy/(a^2+y^2)^{3/2}$ .

Yeh chapter ki neenv (foundation) hai — aage Gauss's law, potential, dipole sab isi pe khade hain. 80/20 funda: bas teen cheez pakad lo — field = force/charge, lines ka matlab, aur vector superposition. Exam ke 80% questions inhi se ban jaate hain.

Electric field — definition, field lines, superposition

1. Definition

Deriving the field of a point charge — from scratch

2. Field lines (Dual coding)

3. Superposition

4. Worked examples

5. Common mistakes

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Electromagnetism

Connections