Intuition The big pictureTwo charges sitting in space feel each other without touching. The closer they are, the harder they push (or pull). The more charge each has, the stronger the effect. Coulomb's law is just the exact arithmetic of that feeling — and it has the same skeleton as Newton's gravity, except charge can be positive or negative, so the electric force can both pull and push , while gravity can only ever pull .
The electrostatic force between two point charges q 1 q_1 q 1 q 2 q_2 q 2 r r r F = k ∣ q 1 q 2 ∣ r 2 F = k\,\frac{|q_1\,q_2|}{r^2} F = k r 2 ∣ q 1 q 2 ∣ k = 1 4 π ε 0 ≈ = = 8.99 × 10 9 = = N⋅m 2 / C 2 k = \dfrac{1}{4\pi\varepsilon_0} \approx ==8.99\times10^{9}==\ \text{N·m}^2/\text{C}^2 k = 4 π ε 0 1 ≈== 8.99 × 1 0 9 == N⋅m 2 / C 2 ε 0 = 8.85 × 10 − 12 C 2 / N⋅m 2 \varepsilon_0 = 8.85\times10^{-12}\ \text{C}^2/\text{N·m}^2 ε 0 = 8.85 × 1 0 − 12 C 2 / N⋅m 2 permittivity of free space .
Like charges repel , unlike charges attract .
The unit of charge is the coulomb (C) . A coulomb is huge — the charge on one electron is only e = 1.6 × 10 − 19 C e = 1.6\times10^{-19}\,\text{C} e = 1.6 × 1 0 − 19 C
We don't "derive" Coulomb's law from something deeper (it is an experimental law), but we can argue why each piece must look the way it does .
1 / r 2 1/r^2 1/ r 2 r r r 4 π r 2 4\pi r^2 4 π r 2 r r r 1 / ( 4 π r 2 ) 1/(4\pi r^2) 1/ ( 4 π r 2 ) That 4 π 4\pi 4 π k = 1 / 4 π ε 0 k = 1/4\pi\varepsilon_0 k = 1/4 π ε 0 — it makes later equations (Gauss's law) clean.
Intuition Why proportional toq 1 q 2 q_1 q_2 q 1 q 2 q 1 q_1 q 1 two copies of the same source, each pulling with the original force, so total doubles. Same for q 2 q_2 q 2 F ∝ q 1 F\propto q_1 F ∝ q 1 F ∝ q 2 F\propto q_2 F ∝ q 2 F ∝ q 1 q 2 F\propto q_1 q_2 F ∝ q 1 q 2
By Newton's third law, F ⃗ 12 = − F ⃗ 21 \vec F_{12} = -\vec F_{21} F 12 = − F 21
Feature
Coulomb (electric)
Newton (gravity)
Source
charge q q q
mass m m m
Constant
k = 8.99 × 10 9 k=8.99\times10^9 k = 8.99 × 1 0 9 G = 6.67 × 10 − 11 G=6.67\times10^{-11} G = 6.67 × 1 0 − 11
Distance law
1 / r 2 1/r^2 1/ r 2 1 / r 2 1/r^2 1/ r 2
Sign
± \pm ± or repelalways attract
Strength
enormous feeble
Can be shielded?
yes
no
For two electrons, take the ratio of electric repulsion to gravitational attraction:
F e F g = k e 2 / r 2 G m e 2 / r 2 = k e 2 G m e 2 \frac{F_e}{F_g} = \frac{k\,e^2/r^2}{G\,m_e^2/r^2} = \frac{k\,e^2}{G\,m_e^2} F g F e = G m e 2 / r 2 k e 2 / r 2 = G m e 2 k e 2
r 2 r^2 r 2 identical 1 / r 2 1/r^2 1/ r 2
Plugging numbers (e = 1.6 × 10 − 19 e=1.6\times10^{-19} e = 1.6 × 1 0 − 19 m e = 9.11 × 10 − 31 m_e=9.11\times10^{-31} m e = 9.11 × 1 0 − 31 F e F g ≈ 4.2 × 10 42 \frac{F_e}{F_g} \approx 4.2\times10^{42} F g F e ≈ 4.2 × 1 0 42
Worked example Reading that numberElectric force between two electrons is about 10 42 10^{42} 1 0 42 Why this matters: gravity only wins on cosmic scales because big objects are electrically neutral (equal + and −), so their electric forces cancel, leaving feeble gravity to dominate.
Worked example Example 1 — basic forceTwo charges q 1 = + 3 μ C q_1=+3\,\mu\text{C} q 1 = + 3 μ C q 2 = − 2 μ C q_2=-2\,\mu\text{C} q 2 = − 2 μ C 0.1 m 0.1\,\text{m} 0.1 m
Step 1: Convert: q 1 = 3 × 10 − 6 q_1=3\times10^{-6} q 1 = 3 × 1 0 − 6 q 2 = 2 × 10 − 6 q_2=2\times10^{-6} q 2 = 2 × 1 0 − 6 Why? Coulomb's law uses SI base units (C, m).
Step 2: F = ( 8.99 × 10 9 ) ( 3 × 10 − 6 ) ( 2 × 10 − 6 ) ( 0.1 ) 2 F = (8.99\times10^9)\dfrac{(3\times10^{-6})(2\times10^{-6})}{(0.1)^2} F = ( 8.99 × 1 0 9 ) ( 0.1 ) 2 ( 3 × 1 0 − 6 ) ( 2 × 1 0 − 6 ) Why magnitudes? We compute size first, decide direction from signs.
Step 3: F = 8.99 × 10 9 × 6 × 10 − 12 0.01 = 8.99 × 10 9 × 6 × 10 − 10 = 5.39 N F = 8.99\times10^9 \times \dfrac{6\times10^{-12}}{0.01} = 8.99\times10^9\times6\times10^{-10} = 5.39\ \text{N} F = 8.99 × 1 0 9 × 0.01 6 × 1 0 − 12 = 8.99 × 1 0 9 × 6 × 1 0 − 10 = 5.39 N
Step 4: Signs opposite → attractive , 5.39 5.39 5.39
Worked example Example 2 — forecast then verifyForecast: if I halve the distance, what happens to the force?
Since F ∝ 1 / r 2 F\propto 1/r^2 F ∝ 1/ r 2 r r r 1 / ( 1 / 2 ) 2 = 4 1/(1/2)^2 = 4 1/ ( 1/2 ) 2 = 4 Predict ×4.
Verify: With r = 0.05 m r=0.05\,\text{m} r = 0.05 m F = 8.99 × 10 9 × 6 × 10 − 12 0.0025 = 21.6 N F = 8.99\times10^9\times\dfrac{6\times10^{-12}}{0.0025} = 21.6\ \text{N} F = 8.99 × 1 0 9 × 0.0025 6 × 1 0 − 12 = 21.6 N 5.39 × 4 = 21.6 5.39\times4 = 21.6 5.39 × 4 = 21.6
Worked example Example 3 — comparisonTwo protons in a nucleus are r = 1 × 10 − 15 r=1\times10^{-15} r = 1 × 1 0 − 15 F = 8.99 × 10 9 × ( 1.6 × 10 − 19 ) 2 ( 10 − 15 ) 2 = 8.99 × 10 9 × 2.56 × 10 − 38 10 − 30 ≈ 230 N F = 8.99\times10^9 \times \dfrac{(1.6\times10^{-19})^2}{(10^{-15})^2} = 8.99\times10^9\times \dfrac{2.56\times10^{-38}}{10^{-30}} \approx 230\ \text{N} F = 8.99 × 1 0 9 × ( 1 0 − 15 ) 2 ( 1.6 × 1 0 − 19 ) 2 = 8.99 × 1 0 9 × 1 0 − 30 2.56 × 1 0 − 38 ≈ 230 N Why striking: 230 N on a subatomic particle is colossal — this is why a strong nuclear force is needed to hold the nucleus together against Coulomb repulsion.
Common mistake "Force doubles when distance doubles is halved? No — inverse, not inverse-square confusion"Wrong belief: doubling r r r F F F Why it feels right: lots of everyday things are linear ("twice as far → half as strong"). Fix: it's 1 / r 2 1/r^2 1/ r 2 r r r 1 / 4 1/4 1/4 1 / 2 1/2 1/2
Common mistake "Plug in microcoulombs directly"Wrong: using 3 3 3 3 × 10 − 6 3\times10^{-6} 3 × 1 0 − 6 Why it feels right: the number "3 µC" looks clean. Fix: convert everything to SI (C, m) before substituting.
Common mistake "Sign the magnitude formula"Wrong: writing F = k q 1 q 2 / r 2 F=k q_1q_2/r^2 F = k q 1 q 2 / r 2 Why it feels right: the vector form does use signs. Fix: in the scalar formula use ∣ q 1 q 2 ∣ |q_1q_2| ∣ q 1 q 2 ∣ separately from the signs. Only the vector form carries the sign.
Common mistake "Coulomb's law works for any charged blobs"Why it feels right: it's so general-looking. Fix: it's exact only for point charges (or spherically symmetric charges, treated as if all charge sits at the centre). For extended shapes you integrate.
Recall Feynman: explain to a 12-year-oldImagine two magnets, but for electricity . Each charged thing has an invisible reach. The reach gets weaker fast as you move away — move twice as far and it's four times weaker (not just two). Plus-and-plus push apart, plus-and-minus snap together. Gravity works the same way with weight instead of charge, but gravity only ever pulls and is unbelievably weaker — electricity between two tiny specks is trillions of trillions of times stronger than their gravity. The only reason we feel gravity at all is that big objects have equal plus and minus, so their electricity cancels out.
Mnemonic Remember the form"k Q Q over r-squared" — sing it like "cookies over our squad."
For strength order: C oulomb is C olossal, G ravity is G entle (and k ∼ 10 9 k\!\sim\!10^9 k ∼ 1 0 9 G ∼ 10 − 11 G\!\sim\!10^{-11} G ∼ 1 0 − 11
What does Coulomb's law state for two point charges? F = k ∣ q 1 q 2 ∣ / r 2 F = k|q_1q_2|/r^2 F = k ∣ q 1 q 2 ∣/ r 2 , directed along the line joining them; like charges repel, unlike attract.
Value of k = 1 / 4 π ε 0 k=1/4\pi\varepsilon_0 k = 1/4 π ε 0 ≈ 8.99 × 10 9 N⋅m 2 / C 2 \approx 8.99\times10^9\ \text{N·m}^2/\text{C}^2 ≈ 8.99 × 1 0 9 N⋅m 2 / C 2 .
Value and meaning of ε 0 \varepsilon_0 ε 0 8.85 × 10 − 12 C 2 / N⋅m 2 8.85\times10^{-12}\ \text{C}^2/\text{N·m}^2 8.85 × 1 0 − 12 C 2 / N⋅m 2 , the permittivity of free space.
If distance doubles, how does Coulomb force change? Drops to one quarter (inverse-square).
Why does r r r F e / F g F_e/F_g F e / F g Both forces
∝ 1 / r 2 \propto 1/r^2 ∝ 1/ r 2 , so distance divides out.
Approx ratio of electric to gravitational force between two electrons? About
4 × 10 42 4\times10^{42} 4 × 1 0 42 .
Key difference between Coulomb and gravity in sign? Charge can be ± so electric force attracts OR repels; gravity always attracts.
When do you use signed charges vs magnitudes? Signed charges in the vector form (sign sets direction); magnitudes in the scalar magnitude formula.
Why does gravity dominate at large scales despite being weaker? Large bodies are electrically neutral (+ and − cancel), leaving only gravity.
Why the 4 π 4\pi 4 π k = 1 / 4 π ε 0 k=1/4\pi\varepsilon_0 k = 1/4 π ε 0 It comes from influence spreading over a sphere's area
4 π r 2 4\pi r^2 4 π r 2 ; makes Gauss's law clean.
Coulomb's law F = k q1 q2 / r^2
Spreading over sphere 4 pi r^2
Two copies of source argument
Signed charges give attract or repel
Newton's third law F12 = -F21
Newton's gravity F = G m1 m2 / r^2
Intuition Hinglish mein samjhoDekho, Coulomb's law bilkul simple idea hai: do charges aapas mein force lagate hain bina chue. Formula hai F = k q 1 q 2 / r 2 F = k\,q_1 q_2 / r^2 F = k q 1 q 2 / r 2 square ke inversely proportional. Matlab agar distance double kar do, to force half nahi, balki one-fourth ho jaati hai. Yeh inverse-square wala part bahut students galat karte hain, isliye dhyan se.
Direction ka rule easy hai: same sign (plus-plus ya minus-minus) wale repel karte hain, opposite sign wale attract karte hain. Numbers nikaalte waqt hamesha SI units mein convert karo — microcoulomb ko × 10 − 6 \times 10^{-6} × 1 0 − 6 ∣ q 1 q 2 ∣ |q_1 q_2| ∣ q 1 q 2 ∣
Ab gravity se comparison: Newton ka gravity law bhi exactly same shape ka hai — G m 1 m 2 / r 2 G\,m_1 m_2/r^2 G m 1 m 2 / r 2 1 / r 2 1/r^2 1/ r 2 10 42 10^{42} 1 0 42
Exam tip (80/20): bas teen cheezein pakki karo — (1) F = k q 1 q 2 / r 2 F=kq_1q_2/r^2 F = k q 1 q 2 / r 2 k = 9 × 10 9 k=9\times10^9 k = 9 × 1 0 9
