# A solenoid where radius length?

1. Dec 5, 2004

### infamous_Q

i've found a lot of equations and relations about solenoids where the lengh tis always much greater than the radius, but what if the radius was much greater than the length?

PS. also does anyone know an equation that could help me figure out the magnetic field strength outside a solenoid? or any equations relating to an electric motor would be helpful too.

thanx!

2. Dec 7, 2004

### cepheid

Staff Emeritus
I guess in principle the magnetic field of a steady current can always be calculated from the Biot-Savart Law:

$$\mathbf{B(r)} = \frac{\mu_0}{4\pi}I\int{\frac{d\mathbf{l'} \times \mathbf{\hat{r}}}{r^2}}$$

However, the integration may not always be...nice. Or even possible. In my electromagnetism text, the example in which the magnetic field directly above the centre of just one circular loop of current-carrying wire was laborious enough. The field of a solenoid was deduced much more easily using Ampere's law:

$$\oint{\mathbf{B}\cdot d\mathbf{l} } = \mu_0 I_{enc}$$

This requires the use of appropriate Amperian loops in various regions...if you haven't encountered these equations from magnetostatics before...don't try to use them without learning more first about them.

As for a solenoid of much greater "girth" than length...it is far from "ideal" in the sense that the field will be less uniform inside and decidedly non-zero outside (a straight field inside and zero outside is the ideal achieved by an infinite solenoid that "very long" solenoids attempt to approximate). I don't have any specific formulas describing this field offhand.

3. Dec 7, 2004

### imabug

My first guess would be to treat it as a loop or coil rather than a solenoid. But then it's been a while since I've tackled these kinds of problems.

4. Dec 8, 2004

### James R

I wrote a paper on this a few years ago, with an exact solution for any kind of solenoid. It turns out that the solutions are quite complicated, and involve modified Bessel functions.

The standard first-year approximation for the field inside a solenoid, provided it is "long" and you're not looking too near the "walls" is:

$$B = \mu_0 n i$$

everywhere inside the solenoid, where $$n$$ is the number of turns per unit length, and $$i$$ is the current.

5. Jan 22, 2008

### katabatic

but how long is how long

is there a ratio between the two, length and radius, that will give a definition of the size.

6. Jan 22, 2008

### katabatic

Dimensions of a Solenoid

In the the equation describing a solenoid, B=unI, it does not specify a length or radius and on things i have read it says that the radius just has to be long compared to the radius. i was wondering if there is a better guideline like a ratio to explain this.