# Straight wire inductance vs wire radius

• supernano
In summary, the relationship between the wire radius and inductance is inverse, meaning that as the wire gets thinner, the inductance increases. This is due to the coupling between filaments of current within the wire, where thinner wires have higher coupling and thus higher inductance. Rosa's derivation applies to DC currents, and the concept can be visualized using Baluncore's example of filaments within the wire. This understanding was significant in the work by Rosa and his team at NBS.

#### supernano

TL;DR Summary
I am looking for an intuitive explanation to why the inductance of a straight wire is larger for thinner wires.
I know that the whole topic of inductance in a straight wire is complicated (and has led to some heated discussions in this forum ). I followed Rosa's derivation and can see that it leads to an inverse relation of the inductance to the wire radius, and from what could understand, the point is that with thinner wires there is more "space" between the edge of the wire and infinity to integrate across. Is that it, or does someone have a better intuitive explanation for this relationship?

Start by thinking of say 6 parallel filaments on the surface of the wire.

The filaments of current flowing on the thin wire are close together, so their magnetic fields have good coupling. On thicker wires, the individual filaments are more separated, so are less well coupled.

Then increase the number of filaments until you are thinking of a current sheet on the surface of a round wire.

• DaveE
Baluncore said:
Start by thinking of say 6 parallel filaments on the surface of the wire.

The filaments of current flowing on the thin wire are close together, so their magnetic fields have good coupling. On thicker wires, the individual filaments are more separated, so are less well coupled.

Then increase the number of filaments until you are thinking of a current sheet on the surface of a round wire.
Thanks @Baluncore, so I thought about this as well and it makes sense for high frequency signals where the skin depth is much smaller than the radius of the wire.. but from what I understand, Rosa's derivation applies to DC currents, which is what makes it less intuitive to me

supernano said:
.. but from what I understand, Rosa's derivation applies to DC currents, which is what makes it less intuitive to me
OK, so add a central filament, to the six peripheral filaments, making seven. Allocate one seventh of the sectional area to each filament. Place the filaments at the geometric mean of the sub-area they represent. The concept then fits the DC model, and the exact same logic follows. As the wire diameter is increased, the coupling between the filaments is reduced.

• supernano
Baluncore's visualization of filaments of current within the wire leads us in the right direction. The mutual inductance between filaments increases as the wire diameter shrinks. (If the filament separation and total current in the wire are both held constant, then the current per filament increases, increasing the magnetic coupling.)

## 1. What is straight wire inductance?

Straight wire inductance refers to the amount of electromagnetic energy stored in a straight wire when an electric current flows through it. It is a measure of the wire's ability to produce a magnetic field.

## 2. How does wire radius affect inductance?

The wire radius has a direct impact on the amount of inductance in a straight wire. A larger wire radius results in a higher inductance, while a smaller wire radius results in a lower inductance. This is because a larger wire can store more energy and produce a stronger magnetic field.

## 3. What is the formula for calculating straight wire inductance?

The formula for calculating straight wire inductance is L = (μ0 * N^2 * A) / l, where L is the inductance in henries, μ0 is the permeability of free space, N is the number of turns in the wire, A is the cross-sectional area of the wire, and l is the length of the wire.

## 4. How does the number of turns in a wire affect its inductance?

The number of turns in a wire also has a direct impact on its inductance. As the number of turns increases, the inductance also increases. This is because each turn of the wire adds to the overall magnetic field and increases the wire's ability to store energy.

## 5. Can the wire material affect its inductance?

Yes, the wire material can affect its inductance. Different materials have different electrical properties, such as conductivity and resistivity, which can impact the wire's ability to produce a magnetic field and, therefore, its inductance. For example, copper is a highly conductive material and is often used in wires for its low resistance and high inductance.