Difficulty: Intermediate-Advanced | Prerequisites: Gauss' law fundamentals, cylindrical Gaussian surfaces, superposition of electric fields, line charge result E = λ/(2πε₀r).
This extends Gauss' law to cylindrical geometry: hollow and solid cylinders, coaxial conductors, and line charges. These geometries appear constantly in real physics and engineering (coaxial cables, charged pipes, wire fields). The key new skill is handling regions where the charge distribution has a hole (a hollow core), and learning to combine Gauss' law results with superposition when symmetry breaks down. If you are comfortable with the infinite-line-charge result and the pillbox method for planes, you are ready for this material.
A hollow non-conducting cylinder with uniform charge density ρ produces a field inside its walls that depends on (r² - a²)/r, where a is the inner radius. Wrapping a metal tube around such a cylinder forces charge onto the metal's inner surface to cancel the enclosed charge. When symmetry breaks (e.g. an off-centre line charge inside), you use superposition, treating each source independently with its own Gauss' law result.
Volume charge density (ρ)
Charge per unit volume, measured in C/m³. For a solid or thick-walled cylinder, ρ tells you how much charge is packed into each cubic metre of material. Think of it as the 3D version of surface charge density.
Line charge density (λ)
Charge per unit length, measured in C/m. Used for infinitely long charged wires or lines. Think of it as how much charge sits on each metre of wire.
Coaxial geometry
An inner cylinder (or wire) surrounded by a concentric outer cylinder, sharing the same axis. This is the geometry of coaxial cables and many textbook problems. The symmetry lets you apply Gauss' law cleanly in each region.
Induced surface charge density (σ_c)
The charge per unit area that appears on the inner surface of a conducting shell in response to charge enclosed within it. Gauss' law forces this: a Gaussian surface inside the conductor must enclose zero net charge (since E = 0 there), so the inner surface charge exactly cancels whatever is inside.
The cylinder has inner radius a, outer radius b, and uniform volume charge density ρ.
Choose a Gaussian cylinder of length ℓ and radius r, coaxial with the charged cylinder.
The enclosed charge is only the charge in the annular region from a to r: Q_enc = π(r² - a²)ℓρ.
Gauss' law on the lateral surface: ε₀E(2πrℓ) = π(r² - a²)ℓρ.
Solving: E = (ρ/2ε₀) × (r² - a²)/r, directed radially outward (for ρ > 0).
Note: at r = a, this gives E = 0 (no charge enclosed). At r = b, it gives the field at the outer surface of the cylinder.
All the charge is enclosed: Q_enc = π(b² - a²)ℓρ.
The result looks like a line charge with λ_eff = π(b² - a²)ρ: E = π(b² - a²)ρ / (2πε₀r) = (b² - a²)ρ / (2ε₀r).
A Gaussian cylinder inside the metal (c < r < d) must enclose zero net charge (since E = 0 in the metal).
Therefore: Q_on_inner_surface + Q_from_cylinder = 0.
Q_from_cylinder = π(b² - a²)ℓρ, so the charge on the metal's inner surface is Q_in = -π(b² - a²)ℓρ.
Since Q_in = 2πcℓσ_c, the induced surface charge density is:
σ_c = -(b² - a²)ρ / (2c)
The negative sign confirms that the inner surface of the metal carries charge opposite in sign to the non-conducting cylinder's charge.
On the outer surface (r = d): since the metal has zero net charge, σ_d = -σ_c × (c/d), ensuring total induced charge sums to zero.
When a line charge λ is placed at an off-centre position (x = d, where d < a) inside the hollow core:
The system no longer has full cylindrical symmetry about any single axis.
You cannot apply Gauss' law to the combined system to find E directly.
Instead, use superposition: compute the field from the line charge alone and the field from the cylinder alone, then add them as vectors.
At point (-b, 0), on the outer surface opposite the line charge:
Field from line charge: E_λ = -λ / [2πε₀(b + d)] x̂ (the distance from the line to this point is b + d).
Field from cylinder: E_cyl = -π(b² - a²)ρ / (2πε₀b) x̂ (as if all the cylinder's charge were on the axis, at distance b).
Total: E = (1/2πε₀) × [-λ/(b + d) - π(b² - a²)ρ/b] x̂
At point (0, b), on the outer surface above the axis:
The line charge is at (d, 0), so the distance to (0, b) is √(b² + d²).
The field from the line charge has both x and y components (it points away from the line at position (d, 0)).
The field from the cylinder points in the +y direction (radially outward from the cylinder's axis at the origin).
You must decompose the line-charge field into x̂ and ŷ components using geometry:
x-component: -λd / [2πε₀(b² + d²)]
y-component: +λb / [2πε₀(b² + d²)]
Add the cylinder's y-component: π(b² - a²)ρ / (2πε₀b)
Combine for total E at (0, b).
Worked numerical example (a = 3 cm, b = 5 cm, ρ = +4 μC/m³, d = 2 cm, λ = -0.1 μC/m):
At (-b, 0): E ≈ 18,459 N/C in the +x direction.
At (0, b): E ≈ (12,402 x̂ - 23,774 ŷ) N/C.
Quantity | Formula | Conditions |
|---|---|---|
Field inside hollow cylinder wall | E = (ρ/2ε₀)(r² - a²)/r | a < r < b |
Field outside cylinder | E = (b² - a²)ρ/(2ε₀r) | r > b |
Induced σ on metal inner surface | σ_c = -(b² - a²)ρ/(2c) | Uncharged metal tube at radius c |
Line charge field | E = λ/(2πε₀r) | Distance r from line |
Distance for off-centre geometry | Use Pythagorean theorem for r to off-axis sources | — |
Coaxial cables (the cable that carries your TV or internet signal) are precisely this geometry: a central wire carrying current, surrounded by an insulating layer, surrounded by a metal shield. Understanding how charge distributes between the inner conductor and outer shield, and why the field outside the cable is zero, comes directly from the Gauss' law analysis practised here. The off-centre line-charge problem mirrors real engineering situations where cables are not perfectly centred inside their shielding.
Students often forget to subtract a² when computing enclosed charge for a hollow cylinder. The charge starts at r = a, not r = 0. Writing Q_enc = πr²ℓρ instead of π(r² - a²)ℓρ is one of the most common errors.
When the problem loses symmetry (off-centre charge), students try to force Gauss' law to give E directly. It cannot. Superposition is the correct approach: treat each source with its own symmetric Gauss' law result, then add vectors.
Students sometimes assume the field from a cylinder at an external point must point toward or away from the off-centre line charge. The cylinder's field still points radially away from the cylinder's own axis, not toward the line charge. Each source generates its field as if the other were not there.
Forgetting to account for both x and y components when the field point is not along one of the axes of symmetry.
⚠️ The hollow-cylinder field formula E = (ρ/2ε₀)(r² - a²)/r is a standard derivation question. Be able to produce it from scratch using Gauss' law, not just recall it.
⚠️ Induced charge on a concentric metal shell is a very common exam problem. The logic is always the same: E = 0 inside the metal forces Q_enc = 0 for a Gaussian surface in the conductor.
⚠️ Superposition problems (off-centre charges, combined geometries) test whether you understand that Gauss' law gives you each source's field independently, and you must add them as vectors.
⚠️ When computing fields at points that are not along a principal axis, expect to decompose into x and y components. Practise the geometry of finding distances and angles to off-axis sources.
Fill in the blank: The enclosed charge for a Gaussian cylinder of radius r inside a hollow cylinder (inner radius a, outer radius b) is Q_enc = ______.
True or false: The field from a uniformly charged hollow cylinder, at a point outside the cylinder, is identical to the field from a line charge with the same total charge per unit length.
True or false: If a line charge is placed off-centre inside a hollow cylinder, you can still use Gauss' law directly to find the total electric field.
Fill in the blank: The induced surface charge density on the inner surface of an uncharged metal tube surrounding a charged cylinder is σ_c = ______.
True or false: At a point on the outer surface of the cylinder along the y-axis, the field from an off-centre line charge on the x-axis has only a y-component.
Answers: 1. π(r² - a²)ℓρ. 2. True. 3. False (no overall symmetry; use superposition). 4. -(b² - a²)ρ/(2c). 5. False (it has both x and y components).
Q: A non-conducting hollow cylinder has inner radius a = 3 cm, outer radius b = 5 cm, and ρ = +4 μC/m³. What is the electric field at r = 4 cm (inside the wall)?
A: E = (ρ/2ε₀)(r² - a²)/r = (4 × 10⁻⁶)/(2 × 8.85 × 10⁻¹²) × (0.04² - 0.03²)/0.04 = (2.26 × 10⁵) × (7 × 10⁻⁴) ≈ 158 N/C, radially outward.
Q: An uncharged metal tube of inner radius c = 6 cm surrounds the cylinder above. What is the induced surface charge density on the metal's inner surface?
A: σ_c = -(b² - a²)ρ/(2c) = -(0.05² - 0.03²)(4 × 10⁻⁶)/(2 × 0.06) = -(1.6 × 10⁻³)(4 × 10⁻⁶)/0.12 = -5.33 × 10⁻⁸ C/m².
Q: Why can you treat the cylinder and the off-centre line charge separately when computing the total field?
A: Because electric fields obey superposition. Each source creates its field as though the other source were absent, and the total field is the vector sum. The cylinder's field depends only on the cylinder's own charge and its axis; the line charge's field depends only on λ and the distance from the line.
Q: At the point (0, b), why does the field from an off-centre line charge (sitting on the x-axis at x = d) have an x-component?
A: The field from a line charge points radially away from (or toward) the line. Since the line is at (d, 0) and the field point is at (0, b), the direction from the line to the point is (-d, b)/√(b² + d²). This direction has a non-zero x-component whenever d ≠ 0.
This material connects to coaxial capacitors (which you will encounter when studying capacitance), where the voltage between inner and outer cylinders is found by integrating the field derived here. The off-centre superposition technique reappears in magnetism (Ampere's law with off-axis wires) and is a prototype for the method of images used in advanced electrostatics. The conductor-shell result (induced charge cancelling enclosed charge) generalises to any conductor geometry and is the principle behind Faraday cages.
Cylindrical Gauss' law, hollow cylinder, coaxial, volume charge density, ρ, line charge density, λ, radial field, induced surface charge, metal shell, conductor, superposition, off-centre charge, vector components, coaxial cable, P212, PHYS 212, University Physics, electromagnetism, Gaussian cylinder