Coulomb's Law And Electric Field

Q: Does Coulomb's law work for charges moving at high speeds?

Coulomb's law is exact only for static or slowly moving point charges. For rapidly moving charges, you must use the full electromagnetic theory. For JEE/NEET electrostatics problems, all charges are static and Coulomb's law applies exactly.

Q: What is meant by the permittivity of a medium?

Permittivity measures how much a medium reduces the electric force between charges compared to vacuum. In a medium with relative permittivity epsilon_r, the Coulomb force is reduced by the factor epsilon_r.

PhysicsElectrostaticsFor JEE aspirants

Coulomb's law gives the electrostatic force between two point charges: $F = \frac{k q _{1} q _{2}}{r ^{2}}$ , where $k = \frac{1}{4 π ε _{0}} \approx 9 \times 1 0^{9} N\cdotm^{2} / C^{2}$ . The electric field at a point is the force per unit positive test charge: $E = F / q_{0}$ , measured in N/C or V/m. Together, these two laws let us compute the force and field of any static charge configuration in JEE and NEET problems, from single point charges to rings, sheets, and solid spheres.

Key Formulas - Quick Reference

Coulomb's force: $F = \frac{1}{4 π ε _{0}} \cdot \frac{q _{1} q _{2}}{r ^{2}}$ , with $k = 9 \times 1 0^{9} N\cdotm^{2} / C^{2}$ in free space.
Quantization of charge: $Q = n e$ , where $e = 1.6 \times 1 0^{- 19} C$ and $n$ is an integer.
Electric field of a point charge: $E = \frac{k q}{r ^{2}}$ , directed away from $+ q$ and toward $- q$ .
Field of an infinite line of charge (linear density $λ$ ): $E = \frac{λ}{2 π ε _{0} r}$ (radial).
Field on the axis of a ring (radius $R$ , charge $Q$ , distance $x$ ): $E = \frac{k Q x}{( R ^{2} + x ^{2} ) ^{3/2}}$ .
Field of an infinite plane sheet (surface density $σ$ ): $E = \frac{σ}{2 ε _{0}}$ (uniform, on both sides).
Thin spherical shell: $E = \frac{k q}{r ^{2}}$ for $r > R$ ; $E = 0$ inside ( $r < R$ ).
Solid non-conducting sphere (uniform volume density): $E = \frac{k q}{r ^{2}}$ outside; $E = \frac{k q r}{R ^{3}}$ inside.

1. Electric Charge and its Properties

Charge is a fundamental property of certain elementary particles like the electron and proton. There are two kinds - positive and negative. Like charges repel and unlike charges attract. The SI unit of charge is the coulomb (C); the CGS unit is the electrostatic unit (esu), with $1 C = 3 \times 1 0^{9} esu$ .

The magnitude of the smallest free charge observed is the elementary charge $e = 1.6 \times 1 0^{- 19} C$ , carried by a proton ( $+ e$ ) or an electron ( $- e$ ).

1.1 Quantization of Charge

The charge on any body is always an integer multiple of the elementary charge:

Q = n e, n = 0, \pm 1, \pm 2, \pm 3, \dots

You cannot have a body with charge $0.5 e$ or $π e$ . Charges of $\pm e /3$ and $\pm 2 e /3$ exist inside protons and neutrons as quark charges, but they are never observed as free charges.

1.2 Conservation of Charge

The total charge of an isolated system is conserved. Charges can move from one body to another, but they cannot be created or destroyed. In a chemical reaction or a nuclear decay, the algebraic sum of charges before and after remains unchanged.

1.3 Additivity of Charge

Charge is a scalar and adds algebraically. If a body has charges $+ 3 μ C$ , $- 2 μ C$ and $+ 5 μ C$ on different parts, the net charge is $+ 6 μ C$ .

1.4 Distribution of Charge on a Conductor

On an isolated charged conductor, the surface charge density is highest where the surface curvature is greatest (sharp points and edges). This is why lightning rods are pointed - the field near a sharp tip becomes very large.

2. Coulomb's Law

The electrostatic force between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them, directed along the line joining them.

In free space (vacuum):

F = \frac{1}{4 π ε _{0}} \cdot \frac{q _{1} q _{2}}{r ^{2}}

where $ε_{0} = 8.854 \times 1 0^{- 12} C^{2} / N\cdotm^{2}$ is the permittivity of free space, and $\frac{1}{4 π ε _{0}} = k \approx 9 \times 1 0^{9} N\cdotm^{2} / C^{2}$ .

In a material medium of relative permittivity $ε_{r}$ (dielectric constant $K$ ):

F = \frac{1}{4 π ε _{0} ε _{r}} \cdot \frac{q _{1} q _{2}}{r ^{2}} = \frac{F _{vacuum}}{ε _{r}}

The permittivity of the medium is $ε = ε_{0} ε_{r}$ .

Vector form

The force on charge $q_{2}$ due to $q_{1}$ , with $r_{12}$ the position vector of $q_{2}$ relative to $q_{1}$ :

F_{12} = \frac{1}{4 π ε _{0}} \cdot \frac{q _{1} q _{2}}{r _{12}^{2}} \overset{r}{^}_{12}

Figure 1: Coulomb's law - equal and opposite forces along the line joining two point charges.

Key features of Coulomb's law

It is a fundamental law based on experimental observation, not derived from anything more basic.
The forces on the two charges form an action-reaction pair - equal in magnitude, opposite in direction, along the line joining them.
The force is always directed along the line joining the two charges (central force).
The electrical force between two point charges is independent of the presence or absence of other charges in the neighbourhood.
It obeys the inverse-square law, exactly like gravitation.

Superposition Principle

The force experienced by a given charge in the field of a number of point charges is the vector sum of the forces exerted on it by each of the other charges considered separately:

F_{net} = F_{1} + F_{2} + F_{3} + \dots

The same rule applies to electric fields (see next section).

Solved Example 1

A particle A carrying charge

2 \times 1 0^{- 6}

C and mass 100 g is fixed at the bottom of a smooth incline of inclination

30°

. Where should another particle B, having the same charge and mass, be placed on the incline so that it remains in equilibrium?

Solution:

Particle B is on the smooth incline. Three forces act on it: (i) gravity $m g$ vertically down, (ii) normal reaction $N$ perpendicular to the incline, and (iii) electrostatic repulsion $F$ from A, directed up along the incline (since both charges are positive and A is below).

Along the incline, for equilibrium of B:

F = m g sin θ

\frac{k q ^{2}}{r ^{2}} = m g sin θ

where $r$ is the distance between A and B along the incline. Solving for $r$ :

r = q \frac{k}{m g sin θ}

Substituting $q = 2 \times 1 0^{- 6}$ C, $k = 9 \times 1 0^{9}$ N·m $^{2}$ /C $^{2}$ , $m = 0.1$ kg, $g = 10$ m/s $^{2}$ , $sin 30° = 0.5$ :

r = (2 \times 1 0^{- 6}) \frac{9 \times 1 0 ^{9}}{0.1 \times 10 \times 0.5} = 2 \times 1 0^{- 6} \times 1.8 \times 1 0^{10} \approx 0.27 m

So B should be placed about 27 cm from A, up along the incline.

Solved Example 2

Two particles A and B carrying charges

8 \times 1 0^{- 6}

C and

- 2 \times 1 0^{- 6}

C respectively are held fixed with a separation of 20 cm. Where should a third charged particle be placed so that it experiences zero net electric force?

Solution:

For the third particle C to experience zero net force, the forces due to A and B must be equal in magnitude and opposite in direction. This means C must lie on the line joining A and B.

Since A and B have opposite signs, both forces would point in the same direction if C were between them - so C cannot lie between A and B. Also, since $∣ q_{A} ∣ > ∣ q_{B} ∣$ , C must be closer to B (the smaller charge) so that the smaller- $r$ dependence compensates for the smaller charge.

Let BC = $x$ , so AC = $0.2 + x$ (with C on the far side of B from A). Let $Q$ be the charge on C. Setting the magnitudes equal:

\frac{k ∣ q _{A} ∣∣ Q ∣}{( 0.2 + x ) ^{2}} = \frac{k ∣ q _{B} ∣∣ Q ∣}{x ^{2}}

\frac{8}{( 0.2 + x ) ^{2}} = \frac{2}{x ^{2}}

Taking square roots:

\frac{2 2}{0.2 + x} = \frac{2}{x} ⟹ 2 x = 0.2 + x ⟹ x = 0.2 m

So C should be placed 20 cm from B, on the side away from A. (The charge $Q$ cancels - any sign works, but the equilibrium is stable only for one sign; for a full stability check both magnitudes and signs must be considered.)

3. Electric Field Intensity

The electric field intensity

E

at a point is the force experienced by a unit positive test charge placed at that point:

E = F / q_{0}

(in the limit

q_{0} \to 0

so the test charge does not disturb the source).

SI unit: newton per coulomb (N/C), equivalent to volt per metre (V/m).

Electric field is a vector quantity. It also obeys the superposition principle:

E_{net} = E_{1} + E_{2} + E_{3} + \dots

3.1 Lines of Force

An electric field line is an imaginary curve drawn so that its tangent at every point gives the direction of the electric field at that point.

Figure 2: Field lines run from a positive charge to a nearby negative charge.

Lines of force originate from positive charges and terminate on negative charges.
Lines of force originate or terminate perpendicular to the surface of a conductor.
The tangent to a field line at any point gives the direction of $E$ at that point.
Field lines never intersect - if they did, the field would have two directions at the point of intersection, which is impossible.
Field lines are continuous curves - they cannot suddenly break, except at charges.
The number of lines per unit area (line density) is proportional to the magnitude of $E$ .

4. Electric Field of Standard Charge Distributions

4.1 Point Charge

At a distance $r$ from a point charge $q$ :

E = \frac{k q}{r ^{2}} = \frac{1}{4 π ε _{0}} \cdot \frac{q}{r ^{2}}

The field points radially outward for $+ q$ , radially inward for $- q$ .

4.2 Uniformly Charged Straight Line (Linear Density $λ$ )

(i) Infinite line: Perpendicular distance $r$ from the wire:

E = \frac{λ}{2 π ε _{0} r} = \frac{2 k λ}{r}

The field is purely radial (perpendicular to the wire).

Figure 3: Field around an infinite line charge is radial and falls as 1/r.

(ii) Finite line, point on perpendicular bisector or on axis: Integrate the contributions from each element $d q = λ d x$ using Coulomb's law - see Solved Example 3 for the axial case.

4.3 Uniformly Charged Ring (Radius $R$ , Total Charge $Q$ )

At a point on the axis, distance $x$ from the centre:

E = \frac{k Q x}{( R ^{2} + x ^{2} ) ^{3/2}}

The field is along the axis. Special cases:

At the centre ( $x = 0$ ): $E = 0$ by symmetry.
Far from the ring ( $x ≫ R$ ): $E \approx k Q / x^{2}$ - the ring behaves like a point charge.
Maximum field on axis occurs at $x = R / 2$ .

Figure 4: Field on the axis of a uniformly charged ring.

4.4 Uniformly Charged Disc (Radius $R$ , Surface Density $σ$ )

At a point on the axis, distance $x$ from the centre:

E = \frac{σ}{2 ε _{0}} [1 - \frac{x}{R ^{2} + x ^{2}}]

In the limit $R \to \infty$ (infinite sheet), this reduces to $E = σ / (2 ε_{0})$ .

4.5 Thin Spherical Shell (Radius $R$ , Total Charge $q$ )

Outside (

r > R

E = \frac{k q}{r ^{2}}

- as if all charge were concentrated at the centre.
On the surface (

r = R

E = \frac{k q}{R ^{2}}

.
Inside (

r < R

E = 0

4.6 Non-conducting Solid Sphere with Uniform Volume Charge Density $ρ$

Total charge $q = ρ \cdot \frac{4}{3} π R^{3}$ .

Outside (

r > R

E = \frac{k q}{r ^{2}}

.
On the surface (

r = R

E = \frac{k q}{R ^{2}}

.
Inside (

r < R

E = \frac{k q r}{R ^{3}} = \frac{ρ r}{3 ε _{0}}

- linear in

r

4.7 Infinite Cylindrical Conductor (Linear Density $λ$ )

Outside (

r > R

E = \frac{λ}{2 π ε _{0} r}

(like an infinite line).
Inside (

r < R

E = 0

(all charge on the outer surface for a conductor).

4.8 Non-conducting Infinite Cylinder with Uniform Volume Density $ρ$

Let the total charge per unit length be $λ = ρ π R^{2}$ .

Outside (

r > R

E = \frac{λ}{2 π ε _{0} r} = \frac{ρ R ^{2}}{2 ε _{0} r}

.
Inside (

r < R

E = \frac{ρ r}{2 ε _{0}}

4.9 Infinite Plane Sheet of Charge (Surface Density $σ$ )

E = \frac{σ}{2 ε _{0}}

Uniform on both sides, directed perpendicular to the sheet (away from a positively charged sheet). The field does not depend on the distance from the sheet.

Figure 5: Field on both sides of an infinite sheet is uniform and equal to sigma over two epsilon zero.

4.10 Two Parallel Infinite Sheets with Opposite Charges ( $+ σ$ and $- σ$ )

Outside the two sheets: fields cancel, so

E_{P} = E_{R} = 0

.
Between the sheets: fields add, so

E_{Q} = \frac{σ}{ε _{0}}

(directed from

+ σ

- σ

This is the standard result used in the parallel-plate capacitor.

Solved Example 3

Find the electric field at a point on the axis of a charged rod of length

L

and linear charge density

λ

. The point is at a distance

a

from the nearer end of the rod.

Figure 6: Element dx of the rod at distance x from field point P.

Solution:

Consider an element of the rod of length $d x$ at a distance $x$ from the field point P, where $x$ ranges from $a$ to $a + L$ . The element carries charge $d q = λ d x$ .

The field due to this element along the axis:

d E = \frac{k d q}{x ^{2}} = \frac{k λ d x}{x ^{2}}

All the contributions point along the axis (same direction), so we integrate directly:

E = \int_{a}^{a + L} \frac{k λ d x}{x ^{2}} = k λ [- \frac{1}{x}]_{a}^{a + L} = k λ (\frac{1}{a} - \frac{1}{a + L})

E = \frac{k λ L}{a ( a + L )}

Directed along the axis, pointing away from the rod (for $λ > 0$ ).

Common Mistakes to Avoid

Watch out

Signs in Coulomb's law: when computing forces, keep signs of charges - a negative product means an attractive force (opposite direction), not a negative magnitude.
Vector nature of $E$ : for a superposition problem, always resolve fields into components before adding. Never add magnitudes unless the fields are collinear.
Field inside a conductor: in electrostatic equilibrium, $E$ is always zero inside a conductor, but the field outside a conductor near its surface is $σ / ε_{0}$ , not $σ / (2 ε_{0})$ .
Ring vs disc formulas: the axial field of a ring is $k Q x / (R^{2} + x^{2})^{3/2}$ ; do not confuse this with the disc formula $\frac{σ}{2 ε _{0}} [1 - \frac{x}{R ^{2} + x ^{2}}]$ .
Infinite sheet vs infinite line: an infinite sheet gives a uniform field independent of distance; an infinite line gives a field that falls as $1/ r$ . Do not use one formula for the other geometry.
Test charge disturbance: the test charge $q_{0}$ in $E = F / q_{0}$ must be small enough not to disturb the source charges - a formality, but conceptually important.

Frequently Asked Questions

Q1. What is the difference between electric force and electric field?

Electric force is what one charge exerts on another and depends on both charges. Electric field is a property of space around a source charge - it exists whether or not a test charge is present. Force $=$ (test charge) $\times$ (field): $F = q_{0} E$ .

Q2. Why does the field inside a uniformly charged spherical shell come out to zero?

By symmetry and Gauss's law, the field inside must be radial and constant on any inner Gaussian sphere. Since no charge is enclosed by that inner sphere, the total flux is zero, forcing $E = 0$ at every interior point. Equivalently, the field contributions from all elements of the shell cancel exactly inside.

Q3. Does Coulomb's law work for charges moving at high speeds?

Coulomb's law is exact only for static (or slowly moving) point charges. For rapidly moving charges, you must use the full electromagnetic theory (retarded potentials, magnetic forces, and radiation effects). For JEE/NEET electrostatics problems, all charges are static and Coulomb's law applies exactly.

Q4. How is Coulomb's law similar to Newton's law of gravitation?

Both are inverse-square, central, act along the line joining the two bodies, and depend on the product of two source quantities (charges vs masses). Differences: gravity is always attractive; electrostatic force can be either, and is roughly $1 0^{36}$ times stronger between an electron and a proton.

Q5. What is meant by the permittivity of a medium?

Permittivity $ε$ measures how much a medium reduces the electric force between charges compared to vacuum. In a medium with relative permittivity $ε_{r}$ (also called dielectric constant $K$ ), the Coulomb force between two charges is reduced by the factor $ε_{r}$ : $F_{medium} = F_{vacuum} / ε_{r}$ .

Q6. Why do electric field lines never cross?

If two field lines crossed at a point, the electric field at that point would have two different directions (the tangent to each line), which is impossible - a field is a single-valued vector at every point. Hence lines never intersect.

Q7. Are Coulomb's law formulas the same for JEE and NEET?

Yes - Coulomb's law, the electric field definition, superposition principle, and the standard results for point-charge, ring, sheet and sphere geometries appear in both JEE Main/Advanced and NEET syllabi. NEET focuses more on conceptual application while JEE Advanced pushes into complex geometries and multi-step integrations, but the underlying formulas are identical.

Previous year questions on Coulomb's Law And Electric Field

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1. Electric Charge and its Properties

1.1 Quantization of Charge

1.2 Conservation of Charge

1.3 Additivity of Charge

1.4 Distribution of Charge on a Conductor

2. Coulomb's Law

Vector form

Key features of Coulomb's law

Superposition Principle

3. Electric Field Intensity

3.1 Lines of Force

4. Electric Field of Standard Charge Distributions

4.1 Point Charge

4.2 Uniformly Charged Straight Line (Linear Density λ)

4.3 Uniformly Charged Ring (Radius R, Total Charge Q)

4.4 Uniformly Charged Disc (Radius R, Surface Density σ)

4.5 Thin Spherical Shell (Radius R, Total Charge q)

4.6 Non-conducting Solid Sphere with Uniform Volume Charge Density ρ

4.7 Infinite Cylindrical Conductor (Linear Density λ)

4.8 Non-conducting Infinite Cylinder with Uniform Volume Density ρ

4.9 Infinite Plane Sheet of Charge (Surface Density σ)

4.10 Two Parallel Infinite Sheets with Opposite Charges (+σ and −σ)

Common Mistakes to Avoid

Frequently Asked Questions

Q1. What is the difference between electric force and electric field?

Q2. Why does the field inside a uniformly charged spherical shell come out to zero?

Q3. Does Coulomb's law work for charges moving at high speeds?

Q4. How is Coulomb's law similar to Newton's law of gravitation?

Q5. What is meant by the permittivity of a medium?

Q6. Why do electric field lines never cross?

Q7. Are Coulomb's law formulas the same for JEE and NEET?

Previous year questions on Coulomb's Law And Electric Field

Ready to master Electrostatics?

4.2 Uniformly Charged Straight Line (Linear Density $λ$ )

4.3 Uniformly Charged Ring (Radius $R$ , Total Charge $Q$ )

4.4 Uniformly Charged Disc (Radius $R$ , Surface Density $σ$ )

4.5 Thin Spherical Shell (Radius $R$ , Total Charge $q$ )

4.6 Non-conducting Solid Sphere with Uniform Volume Charge Density $ρ$

4.7 Infinite Cylindrical Conductor (Linear Density $λ$ )

4.8 Non-conducting Infinite Cylinder with Uniform Volume Density $ρ$

4.9 Infinite Plane Sheet of Charge (Surface Density $σ$ )

4.10 Two Parallel Infinite Sheets with Opposite Charges ( $+ σ$ and $- σ$ )