Part Complete: Calculate the Charge per Length for the Capacitor
Use this premium calculator to determine the charge per unit length of a coaxial capacitor from geometry, dielectric constant, and applied voltage. The tool computes capacitance per length, charge per length, and the electric field at the inner conductor, then visualizes how line charge density changes with voltage.
Coaxial Capacitor Calculator
Calculated Results
Enter your values and click the button to compute the capacitor charge per unit length.
Expert Guide: How to Part Complete Calculate the Charge per Length for the Capacitor
If you need to part complete calculate the charge per length for the capacitor, the key is to identify the capacitor geometry and then connect capacitance to voltage. In many physics and electrical engineering problems, especially those involving coaxial conductors, you are not asked for the total charge on a short isolated component. Instead, you are asked for the charge per unit length, often written as λ and measured in coulombs per meter. This quantity is especially useful when the capacitor is long compared with its radius, because edge effects become small and the electrostatic field is well described by a cylindrical model.
For a coaxial capacitor, the geometry consists of an inner cylindrical conductor of radius a and an outer cylindrical conductor of radius b. Between them is a dielectric with permittivity ε = ε0 εr. Once you know these values and the voltage difference V between the conductors, the charge per length follows from a standard relationship:
Core equation: λ = C′V, where C′ = 2π ε / ln(b/a).
Combining them gives λ = 2π ε V / ln(b/a).
This means the line charge density increases when the voltage increases, increases when the dielectric constant increases, and decreases when the spacing ratio b/a becomes larger. In practical terms, a compact geometry with a high permittivity dielectric stores more charge per meter at the same applied voltage.
What “Charge per Length” Means
Charge per length is simply the amount of charge stored on each meter of the inner conductor, with an equal and opposite charge on the outer conductor. If the line charge density is 20 nC/m, then a 2 m length stores 40 nC on the inner conductor and -40 nC on the outer conductor, assuming ideal electrostatics. This is why the quantity is so useful in cable analysis, RF structures, and introductory electrostatics exercises. It lets you characterize the storage behavior without having to specify the full cable length at the start.
Many students first encounter this concept in Gauss’s law problems. For cylindrical symmetry, the electric field outside the inner conductor depends directly on λ. Then, by integrating the electric field from radius a to radius b, you obtain the potential difference. Rearranging the result gives the capacitance per unit length. That is why λ, V, and C′ are tightly linked.
Step by Step Method
- Identify the geometry as a coaxial capacitor or another long cylindrical capacitor model.
- Measure or convert the inner radius a and outer radius b into meters.
- Determine the dielectric constant εr of the insulating material.
- Compute the absolute permittivity using ε = ε0 εr.
- Calculate capacitance per length using C′ = 2π ε / ln(b/a).
- Multiply by the applied voltage using λ = C′V.
- If needed, multiply by a physical length L to get total charge: Q = λL.
Worked Example
Suppose the inner radius is 1.0 mm, the outer radius is 5.0 mm, the dielectric is polyethylene with εr = 2.25, and the applied voltage is 100 V.
- Convert to meters: a = 0.001 m, b = 0.005 m.
- Compute permittivity: ε = 8.854187817 × 10-12 × 2.25 ≈ 1.992 × 10-11 F/m.
- Compute the logarithmic term: ln(b/a) = ln(5) ≈ 1.609.
- Capacitance per length: C′ = 2π(1.992 × 10-11) / 1.609 ≈ 7.78 × 10-11 F/m.
- Charge per length: λ = C′V ≈ 7.78 × 10-11 × 100 ≈ 7.78 × 10-9 C/m.
The answer is 7.78 nC/m. If the cable segment is 3 m long, then the total charge magnitude on each conductor is approximately 23.3 nC.
Why the Natural Logarithm Appears
In cylindrical geometry, the electric field in the dielectric is inversely proportional to radial distance, E(r) ∝ 1/r. When you integrate 1/r from the inner radius to the outer radius, you obtain a natural logarithm. This is why the ratio of radii matters, not simply the difference between them. If you double both radii while keeping the ratio b/a constant, the capacitance per length stays the same in the ideal coaxial model. That is a subtle but important result in electrostatics.
Real Material Statistics for Relative Permittivity
The dielectric material strongly affects charge storage. Typical values vary with frequency, temperature, and formulation, but standard engineering approximations are still very useful in design and homework calculations.
| Material | Typical Relative Permittivity εr | Common Use | Impact on Charge per Length |
|---|---|---|---|
| Vacuum | 1.000 | Reference medium | Baseline storage capability |
| Dry Air | 1.0006 | Open laboratory setups | Nearly the same as vacuum |
| PTFE | 2.0 to 2.1 | RF coaxial cables | About 2 times vacuum value |
| Polyethylene | 2.25 to 2.30 | General cable insulation | Roughly 2.25 times vacuum value |
| Glass | 4 to 10 | Specialized dielectric structures | Much larger line charge at same voltage |
Because λ is directly proportional to εr, doubling the relative permittivity nearly doubles the charge per meter if voltage and geometry stay fixed. That makes dielectric selection a fundamental design choice.
Comparison of Geometry Effects
Geometry matters through the ratio b/a. A larger ratio means the conductors are effectively farther apart in logarithmic terms, so capacitance per length drops. The table below uses vacuum as the dielectric and shows how geometry changes C′.
| Inner Radius a | Outer Radius b | Ratio b/a | ln(b/a) | Approx. C′ in Vacuum |
|---|---|---|---|---|
| 1 mm | 2 mm | 2 | 0.693 | 80.2 pF/m |
| 1 mm | 5 mm | 5 | 1.609 | 34.6 pF/m |
| 1 mm | 10 mm | 10 | 2.303 | 24.2 pF/m |
| 2 mm | 6 mm | 3 | 1.099 | 50.6 pF/m |
These values show a common engineering tradeoff. A tighter geometry gives more capacitance and more charge per length, but it can also increase electric field stress. If the field becomes too high near the inner conductor, dielectric breakdown becomes a concern. Therefore, when you part complete calculate the charge per length for the capacitor, you should also consider the field strength, not just the stored charge.
Checking the Electric Field
For a coaxial capacitor, the electric field is strongest at the inner conductor surface. The peak field is
E(a) = λ / (2π ε a)
or equivalently, after substituting λ,
E(a) = V / (a ln(b/a)).
This expression is particularly important because it shows the field maximum depends heavily on the inner radius. If the inner conductor is very small, the electric field at its surface can become large even at moderate voltages. In practical design, this may limit operating voltage.
Common Mistakes to Avoid
- Using diameter instead of radius. The formula requires radii. If you have diameters, divide them by two first.
- Forgetting unit conversion. Millimeters must be converted to meters if you want SI-consistent answers.
- Using b – a instead of b/a. The formula depends on the logarithm of the ratio, not the difference.
- Confusing total charge with line charge density. λ is in C/m, while total charge Q is in C.
- Ignoring dielectric effects. If εr changes, capacitance per length and charge per length change proportionally.
When This Calculation Is Used
This calculation appears in electrostatics, transmission line theory, high-voltage cable design, instrumentation, and microwave engineering. Coaxial geometry is one of the most important real-world examples because it provides excellent shielding and a predictable field pattern. In teaching laboratories, charge per length is also a bridge concept between Gauss’s law and capacitance formulas.
For students, the problem often appears as a multi-part question. One part might ask you to derive the electric field with Gauss’s law. The next asks for the potential difference. The final part then asks you to calculate the charge per length for the capacitor. If that is your situation, remember that once C′ is known, the last step is usually straightforward: multiply by V.
How This Calculator Helps
The calculator above automates the most error-prone parts of the process. It converts dimensions to SI units, computes the dielectric permittivity, applies the logarithmic geometry formula, and returns the line charge density in both coulombs per meter and nanocoulombs per meter. It also estimates total charge over a selected reference length, which is useful for comparing textbook answers with practical cable segments.
The graph provides an additional insight: for a fixed geometry and dielectric, the charge per length is a linear function of voltage. That means the plot of λ versus V is a straight line through the origin, as long as the dielectric remains linear and there is no breakdown or strong nonlinear behavior. This is exactly what classical capacitor theory predicts.
Authoritative Learning Resources
If you want to verify constants, derivations, and electrostatics background, consult these reliable educational resources:
- NIST: Vacuum electric permittivity constant
- MIT: Electricity and magnetism study material
- Georgia State University HyperPhysics: Cylindrical capacitor
Final Takeaway
To part complete calculate the charge per length for the capacitor, focus on three inputs: geometry, dielectric, and voltage. For a coaxial capacitor, compute C′ = 2π ε / ln(b/a) and then use λ = C′V. If needed, multiply by the physical cable length to get total charge. This approach is compact, correct, and directly tied to the underlying physics of cylindrical electric fields. Once you understand that relationship, a wide range of capacitor and cable problems become much easier to solve accurately.