Difficulty: Intermediate | Prerequisites: Gauss's law for cylindrical symmetry, linear charge density, natural logarithm in potential integrals.
This topic extends the concentric-geometry pattern from spheres to infinitely long cylinders. The physics is the same (Gauss's law gives the field, then you integrate to get the potential), but the mathematics differs: cylindrical symmetry produces a 1/r field rather than 1/r², so the potential involves a logarithm instead of a 1/r term. This is a common exam setup because it tests whether you can adapt the Gauss's law workflow to a different geometry and handle the ln(r) integral correctly. You should already be comfortable with Gauss's law for cylinders and the concept of linear charge density.
An infinitely long charged insulating cylinder sits inside a concentric cylindrical conducting shell. The electric field falls off as 1/r (not 1/r²), so the potential between surfaces involves a logarithmic function. The conductor still has E = 0 inside, giving a flat potential across its thickness. Combining the inner cylinder's charge density with the shell's linear charge density determines the total enclosed charge at any radius.
Linear charge density (λ)
Charge per unit length along the cylinder axis, in C/m. For a solid cylinder of volume charge density ρ and radius a, λ_inner = ρ · πa². Think of it as squashing all the charge in a thin disc cross-section into a single number.
Volume charge density (ρ)
Charge per unit volume, in C/m³. Related to linear charge density by ρ = λ / (πr²) for a uniformly charged cylinder of radius r.
Enclosed linear charge density (λ_enclosed)
The total λ seen by a Gaussian cylinder at a given radius. Outside everything, λ_enclosed = λ_inner + λ_outer (the shell's own linear charge density).
Potential difference (ΔV)
The difference in electric potential between two points. For cylindrical symmetry, ΔV between two radii involves k · λ_enclosed · ln(r₂/r₁), not a 1/r expression.
Use the cylindrical Gauss's law result:
$$E = 2k \frac{\lambda_{enclosed}}{r}$$
λ_enclosed is the sum of the inner cylinder's linear charge density and the conducting shell's linear charge density: λ_inner + λ_outer.
λ_inner is found from the volume charge density: λ_inner = ρ · πa².
At a point on the y-axis at distance d from the origin, the field points radially. The y-component carries the full magnitude (with sign determined by the net charge).
Points P and R are both outside the shell but at different positions. For example, P at (d, d) and R at (0, d).
The potential difference depends only on the change in radial distance from the axis.
The integration path can be chosen along a convenient direction. If P and R share the same y-coordinate but differ in x, only the x-component of E contributes:
$$\Delta V = -\int_P^R \mathbf{E} \cdot d\mathbf{l}$$
For a path in the x-direction at fixed y, the x-component of E at position (x, y) is E · (x/r), where r = √(x² + y²).
The integral evaluates to:
$$\Delta V = k\lambda_{enclosed} \ln!\left(\frac{r_R}{r_P}\right)$$
where r_R and r_P are the radial distances from the axis to R and P respectively.
Through the conductor (from outer surface c to inner surface b): no change in potential (E = 0).
From inner surface b to insulator surface a: integrate the field produced by the inner cylinder alone (the shell's charge is outside this region):
$$\Delta V = -2k\lambda_{inner} \ln!\left(\frac{a}{b}\right)$$
Note the sign: because a < b, ln(a/b) is negative, and whether ΔV is positive or negative depends on the sign of λ_inner.
If the charge density ρ is positive, the electric field points outward from the axis.
Moving outward from the axis means moving with the field, which decreases the potential.
So V(a) < 0 when the zero of potential is defined on the axis and ρ > 0.
For E to vanish outside the shell, the total enclosed λ must be zero: λ_inner + λ_outer = 0.
Set ρ' · πa² = -λ_outer and solve for ρ'.
This gives a new, typically much larger charge density whose linear charge density exactly cancels the shell's.
Quantity | Expression |
|---|---|
λ from ρ (solid cylinder) | λ = ρ · πa² |
E outside (cylindrical) | E = 2kλ_enclosed / r |
ΔV between two radii | ΔV = –2kλ_enclosed · ln(r₂ / r₁) |
ρ' for zero external field | ρ' = –λ_outer / (πa²) |
Coaxial cables are the engineering equivalent of this geometry: a central conductor, a dielectric gap, and an outer conducting sheath. The potential difference between inner and outer conductors, computed with exactly this logarithmic formula, determines the cable's capacitance per unit length and therefore its signal-carrying characteristics. Every piece of coaxial cable in your home (antenna leads, internet connections) relies on this physics.
Students frequently use the 1/r² (spherical) field formula instead of 1/r (cylindrical). Always check which symmetry you are working with before writing down the field.
The potential integral for cylinders gives a logarithm, not a 1/r term. Mixing these up is one of the most common algebraic errors on exams.
When computing the potential difference between two off-axis points, students sometimes forget that the relevant variable is the radial distance from the axis, not the x- or y-coordinate alone. For a point at (x, y), the radial distance is √(x² + y²).
Students sometimes assume that because the shell has a net charge, the field between the shell and the inner cylinder is affected. It is not: in that region, only the inner cylinder's charge matters (Gauss's law encloses only what is inside your surface).
⚠️ Know the cylindrical field formula cold: E = 2kλ/r. It appears constantly.
⚠️ Be prepared to convert between ρ and λ using λ = ρ · πa². This conversion is tested in nearly every concentric-cylinder problem.
⚠️ Exam questions often ask for the charge density that makes the external field vanish. The method: set λ_inner = –λ_outer and solve for ρ.
⚠️ Path-independence of potential: the potential difference between two points does not depend on the path taken, only on the start and end radial distances. Exam problems test this by giving oblique paths.
True or false: The electric field of an infinitely long charged cylinder falls off as 1/r².
Fill in the blank: The linear charge density of a uniformly charged solid cylinder of radius a and volume charge density ρ is λ = _______.
True or false: The potential difference between two points in cylindrical geometry is proportional to ln(r₂/r₁).
Fill in the blank: Inside the conducting shell material, E = _______ and therefore V is _______.
True or false: To make the electric field zero outside the entire assembly, the inner cylinder's λ must equal the negative of the shell's λ.
Q: An insulating cylinder of radius a = 2.2 cm has ρ = 28 µC/m³. What is its linear charge density?
A: λ = ρ · πa² = 28 × 10⁻⁶ · π · (0.022)² ≈ 4.26 × 10⁻⁸ C/m. Convert a to metres before substituting.
Q: The cylindrical conducting shell has λ_shell = –0.36 µC/m. What is the total enclosed λ at a radius outside the shell?
A: λ_enclosed = λ_inner + λ_shell. Compute λ_inner from ρπa², then add the shell's value. The two may partially cancel, giving a reduced net λ.
Q: Explain why V(c) – V(a) involves only the inner cylinder's charge, not the shell's.
A: Between the inner surface of the shell (r = b) and the insulator surface (r = a), a Gaussian cylinder encloses only the inner cylinder's charge. The shell's charge is outside this region and does not contribute to the field there. Through the shell itself (b to c), E = 0, so that segment adds nothing to the potential difference.
Q: If the zero of potential is defined on the cylinder axis, and ρ > 0, what is the sign of V(a)?
A: V(a) < 0. Moving outward from the axis means moving in the direction of E (for positive charge), which decreases the potential from its zero-reference value on the axis.
This connects to capacitance of coaxial cables: C/L = 2πε₀ / ln(b/a), derived from exactly this potential calculation. It also ties back to Gauss's law for cylinders (the prerequisite topic) and forward to energy stored in the electric field of a cylindrical capacitor. If you later study transmission lines or waveguides, the same geometry and logarithmic potential appear again.
electric potential, cylindrical symmetry, concentric cylinders, coaxial, Gauss's law cylinder, linear charge density lambda, volume charge density rho, logarithmic potential, conducting shell, insulating cylinder, potential difference, E field 1/r, PHYS 212, UIUC, university physics electricity and magnetism