JEE Mains Physics · Electrostatics

Electric Field of Charges, Rods, Rings and Sheets

The field at a point is the force on a unit positive charge placed there; add each charge's kq/r² as a vector, and for rods, rings and sheets let symmetry cancel what it can before you add the rest.

All Electrostatics notes JEE Mains Physics strategy

Why this matters

Thirty-one PYQs, twenty-five of them multiple choice, and three from 2026. Thirteen deal with point charges: null points, charges on a polygon or a cube, and statements about field lines. Eight take the field of a rod, an arc, a ring or a disc, including the point where a ring's axial field peaks. Ten use infinite sheets and line charges, alone or together.

Concept 1 of 3: Field of point charges and null points

A positive charge's field points away from it, a negative charge's towards it, and both fall as 1/r². With several charges, draw each field as an arrow at the point and add the arrows. A null point is where the arrows cancel: between two like charges, or outside two unlike charges on the side of the smaller one. Symmetry also cancels: equal charges at the corners of a regular polygon give no field at its centre.

Definition

$E = \dfrac{kq}{r^{2}}$ , away from a positive charge and towards a negative one. Add the fields as vectors.
Null point, like charges: between them, nearer the smaller, at $x = \dfrac{r}{1 + \sqrt{q_2/q_1}}$ from $q_1$ .
Null point, unlike charges: never between them. It lies outside, beyond the charge of smaller size.
Midway between +q and −q a distance d apart, the two fields point the same way: $E = \dfrac{8kq}{d^{2}}$ , towards −q.
Equal charges at the corners of a regular polygon give $E = 0$ at the centre. If one corner's charge is changed to q′, the field at the centre is that of $q' - q$ alone at that corner.
Field lines start on positive charges and end on negative ones. They never cross, never form closed loops in electrostatics, and are crowded where the field is strong. Inside a conductor the field is zero.

Superposition and the null point of like charges

\vec E = \sum_i \frac{kq_i}{r_i^{2}}\,\hat r_i, \qquad x_{\text{null}} = \frac{r}{1 + \sqrt{q_2/q_1}}

Worked example

(a) Charges

+16\ \mu\text{C}

and

+4\ \mu\text{C}

are 9 cm apart. Where between them is the field zero? (b) Charges

+9\ \mu\text{C}

and

-1\ \mu\text{C}

are 10 cm apart. Where is the field zero?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2023 · 29 January 2023 · Q29Moderate

Example 1 · Electrostatics · Electric Field of Charges, Rods, Rings and Sheets

A point charge

q_{1}= 4q_{0}

is placed at origin. Another point charge

q_{2}= -q_{0}

is placed at

x = 12\text{ }cm

. Charge of proton is

q_{0}

. The proton is placed on

x

axis so that the electrostatic force on the proton is zero. In this situation, the position of the proton from the origin is___

cm

Unlike charges: never between them

Between +q and −q both fields point towards −q, so they add. The null point is outside, beyond the smaller charge. A root between the charges must be rejected.

Add arrows, not numbers

Fields from charges at different places point in different directions. Adding their sizes is right only when they lie along one line and point the same way.

Field lines never cross

At any point the field has one direction, so two lines cannot cross there. Electrostatic field lines also never close on themselves.

Drill 12 more on field of point charges and null points

Concept 2 of 3: Field of rods, arcs, rings and discs

Cut the charge into small pieces and add their fields. Symmetry does most of the work: on a ring's axis, the sideways parts from opposite pieces cancel and only the part along the axis survives. At the centre of a full ring everything cancels. An arc keeps a net field along its line of symmetry, and only the angle it spans matters.

Definition

Rod, on its perpendicular bisector at distance a, with each end at angle θ from the bisector: $E = \dfrac{2k\lambda}{a}\sin\theta$ . A very long rod: $\dfrac{2k\lambda}{a}$ .
Arc of radius R spanning angle φ at the centre: $E = \dfrac{2k\lambda}{R}\sin\dfrac{\varphi}{2}$ , along the line of symmetry. A half ring: $\dfrac{2k\lambda}{R}$ with $\lambda = \dfrac{Q}{\pi R}$ . A full ring: zero.
Ring of charge Q, on its axis at distance z: $E = \dfrac{kQz}{(z^{2} + R^{2})^{3/2}}$ , greatest at $z = \dfrac{R}{\sqrt 2}$ .
Two equal charges a distance 2a apart: on the perpendicular bisector the field is greatest at $\dfrac{a}{\sqrt 2}$ from the midpoint, by the same calculation.
Disc of surface density σ, on its axis: $E = \dfrac{\sigma}{2\varepsilon_0}\left(1 - \dfrac{z}{\sqrt{z^{2} + R^{2}}}\right)$ , which tends to $\dfrac{\sigma}{2\varepsilon_0}$ for a very large disc.

Arc at its centre and ring on its axis

E_{\text{arc}} = \frac{2k\lambda}{R}\sin\frac{\varphi}{2}, \qquad E_{\text{axis}} = \frac{kQz}{(z^{2} + R^{2})^{3/2}}

Worked example

A ring of radius 3 cm carries 5 nC. Find the field on its axis 4 cm from the centre, and where on the axis the field is greatest.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 6 Apr 2026 Shift 1 · Q13Moderate

Example 2 · Electrostatics · Electric Field of Charges, Rods, Rings and Sheets

A thin half ring of radius 35 cm is uniformly charged with a total charge of Q coulomb. If the magnitude of the electric field at center of the half ring is

100\text{ }V/m

, then the value of Q is

\_\_\_\_

nC. (

\epsilon_{0}= 8.85 \times10^{- 12}C^{2}/Nm^{2}

and

\pi= 3.14

)

The arc formula uses half the angle

For an arc spanning φ, the factor is sin(φ/2). A half ring spans 180°, so the factor is sin 90° = 1, not sin 180° = 0.

λ comes from the arc's own length

A half ring of charge Q has λ = Q/πR, not Q/2πR. Use the length that actually carries the charge.

Let symmetry cancel first

On a ring's axis only the axial parts survive. Adding the full kq/r² of every piece overcounts by the factor z/r that the cancelling removes.

Drill 7 more on field of rods, arcs, rings and discs

Concept 3 of 3: Infinite sheets and line charges

An infinite sheet's field does not weaken with distance: move away and the sheet still fills the view. Its field is σ/2ε₀ on each side, pointing away from a positive sheet. With several sheets, work region by region and add each sheet's σ/2ε₀ with its direction. A long line of charge spreads its field over a cylinder, so the field falls as 1/r.

Definition

One sheet: $E = \dfrac{\sigma}{2\varepsilon_0}$ , normal to the sheet, the same at every distance.
Several parallel sheets: in each region, add every sheet's $\dfrac{\sigma}{2\varepsilon_0}$ with its direction. Sheets +σ and −σ give $\dfrac{\sigma}{\varepsilon_0}$ between and zero outside; two +σ sheets give zero between and $\dfrac{\sigma}{\varepsilon_0}$ outside.
Just outside a conductor, with surface density σ on that face: $E = \dfrac{\sigma}{\varepsilon_0}$ .
Two parallel conducting plates: the two outer faces carry equal charges, half of the total; the inner faces carry equal and opposite charges.
A charge between capacitor plates feels the field of both plates. Remove one plate and the field there halves.
Long line charge: $E = \dfrac{\lambda}{2\pi\varepsilon_0 r} = \dfrac{2k\lambda}{r}$ , radially outward for positive λ.

Sheet and line

E_{\text{sheet}} = \frac{\sigma}{2\varepsilon_0}, \qquad E_{\text{line}} = \frac{\lambda}{2\pi\varepsilon_0 r}

Worked example

Two large parallel sheets carry

+3\sigma_0

(left) and

-\sigma_0

(right). Find the field in the three regions.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2024 · 4 April 2024 · Q21Moderate

Example 3 · Electrostatics · Electric Field of Charges, Rods, Rings and Sheets

An infinite plane sheet of charge having uniform surface charge density

+\sigma_{s}C/m^{2}

is placed on

x - y

plane. Another infinitely long line charge having uniform linear charge density

+\lambda_{e}C/m

is placed at

z = 4\text{ }m

plane and parallel to

y

-axis. If the magnitude values

\left| \sigma_{s} \right|= 2\left| \lambda_{e} \right|

then at point

(0,0,2)

, the ratio of magnitudes of electric field values due to sheet charge to that of line charge is

\pi\sqrt{n}:1

. The value of

n

is ______.

One sheet is σ/2ε₀

σ/ε₀ belongs to the surface of a conductor, or to the region between two opposite sheets. A single sheet gives half of that.

A sheet's field does not fall with distance

Only a line charge (1/r) or a point charge (1/r²) weakens with distance. An infinite sheet gives the same field everywhere on one side.

Conducting plates rearrange their charge

Charge given to one plate does not stay on one face. The outer faces always carry equal charges, each half of the total on both plates.

Drill 9 more on infinite sheets and line charges

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Field of point charges and null points
Superposition and the null point of like charges
$\vec E = \sum_i \frac{kq_i}{r_i^{2}}\,\hat r_i, \qquad x_{\text{null}} = \frac{r}{1 + \sqrt{q_2/q_1}}$
Field of rods, arcs, rings and discs
Arc at its centre and ring on its axis
$E_{\text{arc}} = \frac{2k\lambda}{R}\sin\frac{\varphi}{2}, \qquad E_{\text{axis}} = \frac{kQz}{(z^{2} + R^{2})^{3/2}}$
Infinite sheets and line charges
Sheet and line
$E_{\text{sheet}} = \frac{\sigma}{2\varepsilon_0}, \qquad E_{\text{line}} = \frac{\lambda}{2\pi\varepsilon_0 r}$

Watch out for (9)

Unlike charges: never between them
→ Field of point charges and null points
Add arrows, not numbers
→ Field of point charges and null points
Field lines never cross
→ Field of point charges and null points
The arc formula uses half the angle
→ Field of rods, arcs, rings and discs
λ comes from the arc's own length
→ Field of rods, arcs, rings and discs
Let symmetry cancel first
→ Field of rods, arcs, rings and discs
One sheet is σ/2ε₀
→ Infinite sheets and line charges
A sheet's field does not fall with distance
→ Infinite sheets and line charges
Conducting plates rearrange their charge
→ Infinite sheets and line charges

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