Charged Shells & Spheres

Point-charge outside, special inside; and the energy to assemble them.

9 minUpdated 22 October 2026shell theoremelectrostatic self-energy

Lesson at a glanceShow the key points

Outside any uniform spherical distribution: E = Q/4πε₀r², V = Q/4πε₀r.
Thin shell inside: E = 0, V = Q/4πε₀R everywhere.
Solid sphere inside: E ∝ r.
Self-energy: solid sphere 3Q²/20πε₀R; thin shell Q²/8πε₀R.

Coulomb’s law and Newton’s law of gravitation have the same inverse-square form, so every result we proved for the gravity of spheres carries straight over to charged spheres.

Shells and solid spheres

Compare F = q₁q₂/4πε₀r² with F = Gm₁m₂/r²: swap mass for charge and G for 1/4πε₀. For a spherical charge distribution of radius R carrying Q:

	Thin uniform shell	Uniform solid sphere
E outside (r > R)	Q/4πε₀r²	Q/4πε₀r²
E inside (r < R)	0	Qr/4πε₀R³
V outside	Q/4πε₀r	Q/4πε₀r
V inside	Q/4πε₀R (constant)	Q(3R² − r²)/8πε₀R³

Outside, either one acts exactly like a point charge Q at the centre, for both field and potential.
Inside a shell there is no field at all, so the potential is the same everywhere inside, equal to its value on the surface.
Inside a solid sphere the field rises linearly from zero at the centre.

Interactive

Thin shell versus solid sphere

Same total charge Q and radius R. Outside, both act exactly like a point charge at the centre. Inside the shell the field is zero and the potential stays at its surface value; inside the solid sphere E grows in proportion to r and V keeps rising toward the centre, where it is 1.5 times the surface value.

shell E0.00kQ/R²

solid sphere E0.60kQ/R²

energy: shell vs sphere0.50 vs 0.60kQ²/R

Plot

Distance from the centre

r/R

0.60

Energy stored in a charged sphere

The electric potential energy of a charge distribution is the work needed to assemble it, bringing each bit of charge in from infinity.

Uniform solid sphere

Build the sphere layer by layer. When it has reached radius x, with charge density ρ = 3Q/4πR³, it holds q = Qx³/R³ and its surface is at potential

V = \frac{q}{4\pi\varepsilon_0x} = \frac{Q x^2}{4\pi\varepsilon_0 R^3}

The next layer, from x to x + dx, needs dq = 4πx²dx · ρ = 3Qx²dx/R³. Bringing it in costs dW = V dq, and adding all layers:

W = \int_0^R \frac{3Q^2}{4\pi\varepsilon_0 R^6}x^4\,dx = \frac{3Q^2}{20\pi\varepsilon_0 R}

Thin shell

With charge q already on a shell, its potential is q/4πε₀R, so adding dq costs q dq/4πε₀R:

W = \int_0^Q \frac{q\,dq}{4\pi\varepsilon_0 R} = \frac{Q^2}{8\pi\varepsilon_0 R}

Self-energy of charged spheres

U_{\text{sphere}} = \frac{3Q^2}{20\pi\varepsilon_0R}, \qquad U_{\text{shell}} = \frac{Q^2}{8\pi\varepsilon_0R}

Worked example

Shell or ball?

A charge of 1.0 µC is put (a) on a thin shell of radius 10 cm, (b) spread evenly through a ball of radius 10 cm. Find the stored energy in each case.

What to remember

1Outside any uniform spherical distribution: point-charge field and potential.
2Shell: E = 0 inside, V constant = Q/4πε₀R. Solid sphere: E ∝ r inside.
3Self-energy: solid sphere 3Q²/20πε₀R; thin shell Q²/8πε₀R.

Check yourself

Inside a uniformly charged thin shell, the potential…

The same charge on a shell and on a solid sphere of equal radius: which stores more energy?

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

ReadShell theoremWikipedia ReadField of a charged sphereHyperPhysics

PreviousFields from Symmetry Next Conductors & Earthing

Charged Shells & Spheres

Point-charge outside, special inside; and the energy to assemble them.

9 minUpdated 22 October 2026shell theoremelectrostatic self-energy

Lesson at a glanceShow the key points

Outside any uniform spherical distribution: E = Q/4πε₀r², V = Q/4πε₀r.
Thin shell inside: E = 0, V = Q/4πε₀R everywhere.
Solid sphere inside: E ∝ r.
Self-energy: solid sphere 3Q²/20πε₀R; thin shell Q²/8πε₀R.

Coulomb’s law and Newton’s law of gravitation have the same inverse-square form, so every result we proved for the gravity of spheres carries straight over to charged spheres.

Shells and solid spheres

Compare F = q₁q₂/4πε₀r² with F = Gm₁m₂/r²: swap mass for charge and G for 1/4πε₀. For a spherical charge distribution of radius R carrying Q:

	Thin uniform shell	Uniform solid sphere
E outside (r > R)	Q/4πε₀r²	Q/4πε₀r²
E inside (r < R)	0	Qr/4πε₀R³
V outside	Q/4πε₀r	Q/4πε₀r
V inside	Q/4πε₀R (constant)	Q(3R² − r²)/8πε₀R³

Outside, either one acts exactly like a point charge Q at the centre, for both field and potential.
Inside a shell there is no field at all, so the potential is the same everywhere inside, equal to its value on the surface.
Inside a solid sphere the field rises linearly from zero at the centre.

Interactive

Thin shell versus solid sphere

shell E0.00kQ/R²

solid sphere E0.60kQ/R²

energy: shell vs sphere0.50 vs 0.60kQ²/R

Plot

Distance from the centre

r/R

0.60

Energy stored in a charged sphere

The electric potential energy of a charge distribution is the work needed to assemble it, bringing each bit of charge in from infinity.

Uniform solid sphere

Build the sphere layer by layer. When it has reached radius x, with charge density ρ = 3Q/4πR³, it holds q = Qx³/R³ and its surface is at potential

V = \frac{q}{4\pi\varepsilon_0x} = \frac{Q x^2}{4\pi\varepsilon_0 R^3}

The next layer, from x to x + dx, needs dq = 4πx²dx · ρ = 3Qx²dx/R³. Bringing it in costs dW = V dq, and adding all layers:

W = \int_0^R \frac{3Q^2}{4\pi\varepsilon_0 R^6}x^4\,dx = \frac{3Q^2}{20\pi\varepsilon_0 R}

Thin shell

With charge q already on a shell, its potential is q/4πε₀R, so adding dq costs q dq/4πε₀R:

W = \int_0^Q \frac{q\,dq}{4\pi\varepsilon_0 R} = \frac{Q^2}{8\pi\varepsilon_0 R}

Self-energy of charged spheres

U_{\text{sphere}} = \frac{3Q^2}{20\pi\varepsilon_0R}, \qquad U_{\text{shell}} = \frac{Q^2}{8\pi\varepsilon_0R}

Worked example

Shell or ball?

A charge of 1.0 µC is put (a) on a thin shell of radius 10 cm, (b) spread evenly through a ball of radius 10 cm. Find the stored energy in each case.

What to remember

1Outside any uniform spherical distribution: point-charge field and potential.
2Shell: E = 0 inside, V constant = Q/4πε₀R. Solid sphere: E ∝ r inside.
3Self-energy: solid sphere 3Q²/20πε₀R; thin shell Q²/8πε₀R.

Check yourself

Inside a uniformly charged thin shell, the potential…

The same charge on a shell and on a solid sphere of equal radius: which stores more energy?

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

ReadShell theoremWikipedia ReadField of a charged sphereHyperPhysics

PreviousFields from Symmetry Next Conductors & Earthing

Shells and solid spheres#

Energy stored in a charged sphere#

Uniform solid sphere

Thin shell

Check yourself

Keep exploring

Shells and solid spheres#

Energy stored in a charged sphere#

Uniform solid sphere

Thin shell

Check yourself

Keep exploring

Shells and solid spheres

Energy stored in a charged sphere

Shells and solid spheres

Energy stored in a charged sphere