Q.What is the force between two small charged spheres having charges of 2×10−7C and 3×10−7C placed 30cm apart in air?
Concept understanding — Coulomb Force Superposition
Coulomb Force Superposition – From Intuition to Precision
Imagine you're in a room with three friends. Each friend can push or pull you. If two friends push you from the same side, you feel a stronger push — the combined effect. If one pushes from the left and another from the right, you feel the net effect, which might be smaller or even zero if they push equally hard.
This is exactly how electric forces work. When multiple charged particles are present, each one exerts its own force on a given charge. The total force that charge feels is simply the vector sum of all the individual forces — as if each other charge were acting alone, completely ignoring the presence of the rest.
That's the core idea: forces add like arrows, not like numbers.
The Precise Statement
Fnet on q0=∑i=1nFi→0=4πε01∑i=1nri02q0qir^i0
Where:
- q0 is the charge you're calculating the force on
- qi are all other charges (excluding q0 itself)
- ri0 is the distance between qi and q0
- r^i0 is a unit vector pointing from qi to q0 (or away, depending on sign convention — be consistent)
The key point: Each pair of charges interacts independently. The presence of a third charge does not alter the force between the first two. This is what "superposition" means — the forces simply layer on top of each other.
Why This Matters (and a Common Trap)
Never add the magnitudes of forces directly unless all forces are along the same line and in the same direction. Force is a vector — direction matters.
If two forces point in opposite directions, they partially cancel. If they're at right angles, the net force is found using the Pythagorean theorem, not simple addition.
Example: Three charges on a line:
- q1=+2μC at x=0
- q2=−1μC at x=3cm
- q0=+1μC at x=1cm
Step 1: Force from q1 on q0 — both positive, so repulsive. q0 is pushed to the right.
Step 2: Force from q2 on q0 — opposite signs, so attractive. q0 is pulled to the right (toward q2).
Step 3: Both forces point right. Now you add magnitudes: Fnet=F1→0+F2→0.
If q2 were also positive, the force from q2 would push q0 left, and you'd subtract.
The Deeper Reason
Coulomb's law is a linear law — the force is proportional to each charge individually. If you double q1, the force from q1 doubles, but the force from q2 stays the same. This linearity is what makes superposition possible. It's not a coincidence — it's a fundamental property of electromagnetic interactions at the classical level.
Superposition works because electric forces obey a linear inverse-square law. If the force depended on products of three charges (like q0q1q2), superposition would fail. It doesn't — and that's why we can break down any multi-charge problem into a series of two-charge calculations.
How to Use It in Exams
- Draw all charges and label distances.
- For each other charge, sketch the direction of the force on your target charge (like charges repel, opposites attract).
- Write the magnitude of each force using Coulomb's law.
- Resolve into components if forces aren't along the same line.
- Add components separately: Fnet,x=∑Fi,x, same for y, z.
- Combine components to get the net force vector.
In symmetric arrangements (e.g., an equilateral triangle with equal charges), many components cancel. Always check for symmetry before diving into heavy algebra — it can save you minutes.
One Last Check
If you place a test charge q0 at a point and there are 10 other charges around it, you calculate 10 separate Coulomb forces and add them as vectors. That's it. No extra physics, no hidden interactions. The universe, at this level, is beautifully simple: each pair talks only to each other, and you just listen to all the conversations at once.
"Coulomb's law superposition principle examples" and "electrostatics class 12 physics important questions" are frequently searched, both grounded in the Electrostatics chapter of the NCERT/CBSE Class 12 Physics curriculum. Multi-charge force problems using superposition are a near-guaranteed topic in JEE Main and NEET.
Why this formula?
Coulomb Force Superposition — Why the Formula Holds
The principle of superposition for Coulomb forces states that the net electrostatic force on a given charge due to a collection of other charges is the vector sum of the individual forces from each charge, as if the others were absent.
The Key Formula
If we have a charge q0 at position r0, and N other point charges q1,q2,…,qN at positions r1,r2,…,rN, the net force on q0 is:
Fnet=4πε01∑i=1N∣r0−ri∣2q0qir^0i
where r^0i is the unit vector pointing from qi to q0.
Why This Works — The Physical Reasoning
1. Coulomb's Law is a Two-Body Interaction
Coulomb's law describes the force between exactly two point charges. It depends only on:
- The product of their charges (q0qi)
- The inverse square of the distance between them
- The direction along the line joining them
Crucially, the force between q0 and qi does not depend on the presence of any other charges qj.
2. Forces Add as Vectors (Newton's Third Law + Linearity)
Electrostatic forces are real physical forces — they obey Newton's laws. If multiple forces act on the same charge, the net effect is the vector sum of each individual force. This is a fundamental property of forces in classical mechanics.
3. The Electric Field is Linear
A deeper reason: the electric field E obeys superposition. Since F=q0E, and E from multiple sources adds linearly, the force automatically adds linearly.
The electric field at r0 due to qi is:
Ei(r0)=4πε01∣r0−ri∣2qir^0i
Then:
Fnet=q0∑iEi=∑iFi
The Crucial Assumption (Why It's Not Trivial)
Superposition holds because Maxwell's equations are linear in the electric field. If the equations were nonlinear (e.g., if the field depended on E2), then the force from two charges together would not be the sum of the individual forces.
In electrostatics, the electric field satisfies:
∇⋅E=ε0ρ,∇×E=0
Both equations are linear — if E1 and E2 are solutions, then E1+E2 is also a solution. This linearity is the mathematical reason superposition works.
Exam-Relevant Takeaway
| Concept | Why It Holds |
|---|---|
| Superposition of forces | Coulomb force is a two-body interaction; forces add as vectors |
| Superposition of fields | Maxwell's equations are linear in E |
| Net force formula | Fnet=∑Fi — vector sum of individual Coulomb forces |
Never forget: The unit vector r^0i points from the source charge to the test charge — this determines the correct direction of each term.
Quick Example (To Cement the "Why")
Suppose q0=+1μC at the origin, q1=+2μC at (1,0), q2=−2μC at (0,1).
- Force from q1: repulsive, along +x direction
- Force from q2: attractive, along +y direction
The net force is not just the sum of magnitudes — it's the vector sum:
Fnet=F1x^+F2y^
This works because the two forces are independent — q1 doesn't "know" about q2, and vice versa. The superposition principle is simply the statement that this independence holds.
Concept: Coulomb's law for the electrostatic force between two point charges.
The force between two point charges is given by Coulomb's law:
F=kr2∣q1q2∣
where k=9×109N m2/C2 is Coulomb's constant.
Step 1: Identify the given quantities.
q1=2×10−7C, q2=3×10−7C, and r=30cm=0.3m.
Step 2: Substitute into Coulomb's law.
F=9×109×(0.3)2(2×10−7)(3×10−7)=9×109×0.096×10−14
Step 3: Simplify.
F=9×109×9×10−26×10−14=6×10−3N=6mN
Since both charges are positive, the force is repulsive.
The force between the two charged spheres is 6×10−3N or 6mN, repulsive in nature.
Two point charges interact via Coulomb's law: the force is proportional to the product of charges and inversely proportional to the square of their separation. Here F=6×10−3N (repulsive).
Why Coulomb's Law Works Here
When two charged objects are small compared to the distance between them, we can treat them as point charges. The electrostatic force between them follows Coulomb's law, one of the fundamental inverse-square laws in physics. The force is attractive if the charges have opposite signs and repulsive if they have the same sign.
The magnitude of this force depends on three things: how much charge each sphere carries, how far apart they are, and the medium between them. In air (or vacuum), the proportionality constant is Coulomb's constant k=9×109N⋅m2/C2.
F=kr2∣q1q2∣
Step-by-Step Calculation
1. Identify the given quantities
We have:
- First charge: q1=2×10−7C
- Second charge: q2=3×10−7C
- Separation: r=30cm=0.30m
- Medium: air, so k=9×109N⋅m2/C2
Always convert centimeters to meters before substituting into Coulomb's law. The SI unit for distance in this formula is the meter, and mixing units is a common source of error.
2. Substitute into Coulomb's law
F=9×109×(0.30)2(2×10−7)(3×10−7)
3. Simplify the numerator
The product of the charges:
q1q2=2×10−7×3×10−7=6×10−14C2
4. Simplify the denominator
r2=(0.30)2=0.09m2
5. Complete the calculation
F=9×109×0.096×10−14
F=9×109×9×10−26×10−14
F=9×109×96×10−14×102
F=6×109−14+2
F=6×10−3N
6. Interpret the result
Both charges are positive, so the force is repulsive. The magnitude is 6×10−3N or 6mN.
When both charges have the same sign (both positive or both negative), the force pushes them apart. When they have opposite signs, the force pulls them together.
The force between the two charged spheres is 6×10−3N (repulsive).
Concept: Coulomb Force Superposition
The electrostatic force between two point charges is given by Coulomb’s law:
F=kr2∣q1q2∣,k=9×109 N⋅m2/C2
Steps
-
Identify values
q1=2×10−7 C, q2=3×10−7 C, r=30 cm=0.3 m.
-
Apply Coulomb’s law
F=9×109×(0.3)2(2×10−7)(3×10−7)
- Simplify Numerator: 9×109×6×10−14=54×10−5 Denominator: 0.09=9×10−2
F=9×10−254×10−5=6×10−3 N
Final Answer
F=6×10−3 N (repulsive, since both charges are positive).
Here are the most common mistakes students make when solving this classic Coulomb force superposition problem, along with how to avoid each.
1. Forgetting the Vector Nature of Force
The Mistake:
Students often compute the magnitude of the force from each q on Q correctly, but then simply add them as scalars (e.g., Fnet=F1+F2+F3).
Why it’s wrong:
Coulomb force is a vector. Forces from different charges point in different directions. Adding magnitudes directly ignores direction and gives an incorrect (usually larger) result.
How to Avoid:
Always draw a clear diagram showing the direction of each force vector. Use vector addition (component method or symmetry) — never scalar addition.
2. Not Using Symmetry to Simplify
The Mistake:
Students calculate all three force vectors explicitly, resolve into components, and sum — a long, error-prone process.
Why it’s wrong:
It wastes time and increases the chance of algebraic mistakes. The problem has perfect symmetry.
How to Avoid:
Recognize that the three charges are identical and placed at vertices of an equilateral triangle. The centroid is equidistant from all vertices. By symmetry, the three force vectors are equal in magnitude and spaced 120∘ apart. Their vector sum is zero.
Key result: The net force on Q at the centroid is Fnet=0.
3. Incorrect Distance Calculation
The Mistake:
Using l (side length) as the distance between a vertex charge and the centroid.
Why it’s wrong:
The distance from a vertex to the centroid of an equilateral triangle is not l. It is 3l.
How to Avoid:
Memorize or derive:
- Centroid divides the median in ratio 2:1.
- Median length =23l.
- Distance from vertex to centroid =32×median=32⋅23l=3l.
Use r=3l in Coulomb’s law.
4. Sign Confusion in Force Direction
The Mistake:
If Q and q have the same sign, students sometimes draw forces as attractive.
Why it’s wrong:
Like charges repel. All three forces on Q are repulsive and point radially outward from each vertex.
How to Avoid:
Always check: same sign → repulsion (force away from the other charge). Opposite sign → attraction (force toward the other charge). Draw arrows accordingly.
5. Assuming the Net Force is Non-Zero Without Checking
The Mistake:
After computing magnitudes, students assume the forces don’t cancel and proceed to find a non-zero resultant.
Why it’s wrong:
Symmetry guarantees cancellation. The three equal-magnitude vectors at 120∘ to each other always sum to zero.
How to Avoid:
Before doing heavy algebra, pause and check for symmetry. If the configuration is symmetric and all charges are identical, the net force at the center is zero.
Quick Summary Checklist
| Mistake | How to Avoid |
|---|---|
| Scalar addition of forces | Always use vector addition |
| Ignoring symmetry | Use symmetry to simplify first |
| Wrong distance (l instead of l/3) | Derive or memorize centroid distance |
| Wrong force direction (attraction instead of repulsion) | Same sign → repulsion |
| Assuming net force is non-zero | Check symmetry — here it’s zero |
Final takeaway: For this exact problem, the answer is zero — but only if you handle vectors, distances, and directions correctly.
- Higher Secondary (+2 Stage) Examination 2025Set ANNUAL1 markMCQQ.The force acting between two charged particles situated at a distance 'd' is 9N. If the distance between them becomes '3d', the force will be —(a) 1N(b) 3N(c) 6N(d) 27N
›Reveal solutionSolution
Coulomb's force falls off as the inverse square of distance; tripling the separation cuts the force to 1/9th.
Coulomb's law: F=r2kq1q2, so F∝r21.
At separation d: F1=9 N.
At separation 3d: F2=F1×(3dd)2=9×91=1 N.
✓Final answer(a) 1 N.
- Higher Secondary (+2 Stage) Examination 2023Set ANNUAL1 markMCQQ.For free space (vacuum), in the relation F = Kq1q2/r², the value of K will be -(a) 9×10^9 Nm²C^-2(b) (1/9)×10^9 Nm²C^-2(c) 9×10^-9 Nm²C^-2(d) (1/9)×10^-9 Nm²C^-2
›Reveal solutionSolution
Coulomb's constant K equals 1/(4πε₀), and its standard value in SI units (vacuum) is 9×10^9 Nm²C⁻².
Coulomb's law is written F = Kq1q2/r², where K = 1/(4πε₀).
Using ε₀ = 8.85×10⁻¹² C²N⁻¹m⁻² (permittivity of free space):
K = 1/(4π × 8.85×10⁻¹²) ≈ 9×10^9 Nm²C⁻²
This is a fixed, well-known constant for free space/vacuum (and very nearly the same in air).
✓Final answer(a) 9×10^9 Nm²C^-2.
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