Q.A charge 'q' is placed at the midpoint of the line joining two equal charges 'Q'. For what value of 'q' will the entire charge system remain in equilibrium? OR Derive the expression for the torque acting on an electric dipole placed in a uniform electric field.
🔒You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.
🔒 Start your 14-day free trial to unlock the full solution →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.
--- …
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.
--- …
For the whole system to stay in equilibrium each outer charge Q must also feel zero net force, so the pull from the middle charge has to cancel the repulsion between the two Q's, and balancing those two Coulomb forces is what fixes the middle charge's sign and size. …
The middle charge is automatically in equilibrium by symmetry for ANY value of q; the real condition comes from requiring the two end charges Q to also be in equilibrium, which forces q = -Q/4.
Let the two equal charges +Q be placed at points A (x=0) and B (x=2a), with charge q at the midpoint M (x=a).
Equilibrium of q: By symmetry, the two forces on q due to the equal charges Q at A and B (each at distance a) are equal in magnitude and act in opposite directions along the line — so q is automatically in equilibrium at the midpoint, for ANY value or sign of q. This alone does not fix q.
Equilibrium of the whole system: For the SYSTEM to be in equilibrium, the charges at A and B must ALSO be in equilibrium (not just q). Consider the charge Q at A. It experiences:
- A repulsive force from the other charge Q at B (separation 2a): F1=(2a)2kQ2=4a2kQ2, directed away from B (pushing A further left).
- A force from charge q at M (separation a): F2=a2kQq. …
Showing the 12 most recent of 40 on this concept.
- CBSE 2026Set A1 markMCQQ.Coulomb's law is valid for (A) Point charges only (B) Dispersed charges only (C) Both point charges and dispersed charges (D) Neutral particles
›Reveal solutionSolution
Coulomb's law is defined for point charges; extended bodies need integration.
Coulomb's law states F=4πε01r2q1q2, where r is the distance between the charges.
…
- CBSE 2026Set ANNUAL1 markMCQQ.Two spheres carrying charges +6 μC and +9 μC, separated by a distance d, experience a force of repulsion F. When a charge of −3 μC is added to each sphere and distance d is kept the same, the new force of repulsion will be(a) 3F(b) F/9(c) F(d) F/3
›Reveal solutionSolution
New force = F/3, because force is proportional to the product of the charges and only the charges changed, not the distance.
By Coulomb's law, the force between two point charges q1 and q2 separated by a fixed distance d is
F=4πε01d2q1q2
Originally q1=+6 μC and q2=+9 μC, so F∝q1q2=54 (in μC2).
After −3 μC is added to each sphere: …
- CBSE 2026Set ANNUAL1 markMCQQ.Two sphere of charge 2μc and 3μc are located at a distance 20 cm apart in air. The ratio of magnitude of electric forces acting between these spheres will be(a) 1 : 1(b) 2 : 3(c) 3 : 2(d) 4 : 9
›Reveal solutionSolution
The mutual electric force between two charges is an action-reaction pair, so both spheres feel equal magnitude forces regardless of the charge values.
By Coulomb's law the force sphere 1 exerts on sphere 2 has magnitude F = k q1 q2 / r^2, and the force sphere 2 exerts on sphere 1 has the same magnitude k q1 q2 / r^2, just opposite in direction (Newton's third law applies to electrostatic forces just as it do …
- CBSE 2025Set X11 markMCQQ.A point charge q1 exerts a force F on another point charge q2 when placed at a fixed distance. If another point charge q3 is brought near q2, the force on q2 due to q1 :(a) increases(b) decreases(c) may increase or decrease(d) does not change
›Reveal solutionSolution
(d) does not change. By the principle of superposition, the electrostatic force between q1 and q2 is given by Coulomb's law F=4πε01r2q1q2 and depends …
- CBSE 2025Set D1 markMCQQ.The distance between two charges is made half and one of the charges is also halved. The force acting between the two will become as compared to previous value (A) half (B) double (C) thrice (D) none of these
›Reveal solutionSolution
Coulomb force F ∝ q₁q₂/r²; halving one charge (×½) and halving the distance (×4) gives a net factor of 2, so the force doubles.
Coulomb's law:
F=r2kq1q2
Initial force: F=r2kq1q2.
Now one charge becomes q1/2 and the distance becomes r/2:
…
- CBSE 2025Set D1 markMCQQ.On inserting a dielectric material between two positive charges in air, the value of repulsive force will (A) increase (B) decrease (C) remain same (D) become zero
›Reveal solutionSolution
A dielectric weakens the field between charges by a factor K, so the repulsive force falls to F₀/K.
The Coulomb force between two charges in air is F₀ = (1/4πε₀)·q₁q₂/r². Filling the space with a dielectric of relative permittivity K replaces ε₀ with Kε₀:
F = (1/4πKε₀)·q₁q₂/r² = F₀/K
…
- CBSE 2025Set ANNUAL1 markMCQQ.The law governing the force between static electric charges is known as(i) Ampere's law(ii) Ohm's law(iii) Faraday's law(iv) Coulomb's law
›Reveal solutionSolution
The force between two static (point) electric charges is governed by Coulomb's law.
Ampere's law relates a magnetic field to the current producing it, Ohm's law relates current and voltage in a conductor, and Faraday's law deals with electromagnetic induction. None of these describes the force between charges at rest.
…
- CBSE 2024Set IMPROVEMENT1 markMCQQ.On placing dielectric material between two point charges in air, repulsive force between them will —(a) Increase(b) Decrease(c) Remain same(d) Zero
›Reveal solutionSolution
Placing a dielectric between two charges reduces the force between them.
…
- CBSE 2024Set FS1 markMCQQ.Force of 80 Newton works between two point charges placed at a fixed distance apart in air. When these charges are placed at the same distance apart in a dielectric medium, then force of 8 Newton works on it. The dielectric constant of medium will be:(i) K=−10(ii) K=10(iii) K=0.01(iv) K=−0.01
›Reveal solutionSolution
K=FmediumFair=880=10 — option (ii).
Concept. Coulomb's force between two charges at separation r is
Fair=4πε01r2q1q2,Fmedium=4πε0K1r2q1q2.
Placing a dielectric of constant K reduces the force by the factor K. …
- CBSE 2024Set A1 markQ.Match Column 'A' with Column 'B' and write the correct pair. Column 'A' item: 'Electrostatic force'. Column 'B' options:(i) De-Broglie(ii) Maxwell(iii) Ohm(iv) Einstein(v) Coulomb(vi) Lenz(vii) Young.
›Reveal solutionSolution
Electrostatic force is governed by Coulomb's law.
The force of attraction or repulsion between two stationary point charges is called the electrostatic (or Coulomb) force, and its magnitude is given by Coulomb's law:
F=4πε01r2q1q2
…
- CBSE 2024Set ANNUAL1 markMCQQ.Two charged spheres are separated by a distance d, exert a force F on each other. If the charges are doubled and the distance between them is doubled then the force is(a) F(b) F/2(c) F/4(d) 4F
›Reveal solutionSolution
Coulomb's law force scales as (charge product)/(distance)^2; doubling both charges and the distance leaves the force unchanged.
By Coulomb's law, the force between two point charges q1 and q2 separated by distance d is
F=4πϵ01d2q1q2=kd2q1q2
…
- CBSE 2024Set ANNUAL1 markQ.What is the name of the electrical force acting between two charges at rest?
›Reveal solutionSolution
The force between two charges at rest is the electrostatic (Coulomb) force.
The electrical force acting between two charges that are at rest (not moving) is called the electrostatic force, or Coulomb force, since it is governed by Coulomb's law:
F=4πϵ01r2q1q2
…
🎓Unlock everything free for 14 days
- ✓Full step-by-step solutions
- ✓Concept-first explanations
- ✓Methods, shortcuts & mistakes
- ✓PYQ mapping + timed mock tests
Full access for 14 days. No credit card required.