How does the length of a solenoid affect its magnetic field strength?

[ft92]

Homework Statement

Suppose that you want to design a solenoid to control magnetically the central locking system of your car doors. This needs a magnetic induction at the end of the coil of 0.161 T. To be compact enough to fit next to the window winder mechanism, the coil can be only 3.03 cm long and 0.688 cm in diameter. The car battery can supply 8.23 A.

How many turns of wire does the coil need to have if it has air as its core?

Homework Equations

B = μ₀I N / L

The Attempt at a Solution

0.161 = 4π x 10 -7 * 8.23 * N / 0.0303

N = 471.69 turns approximately

I don't know how to take into account the information given about the diameter 0.688 cm.
How can I adjust the formula to get a more accurate result?

 

[gneill]

Is this an open question where you need to research the properties of the materials involved such as the resistivity of the wire, its current carrying capacity, and so on? For example, the resistance of the coil will depend upon resistivity of the wire material, its cross sectional area and total length. The current drawn depends upon the resistance and supply voltage.

Coils can be multi-layered, too, increasing the effective number of turns per unit length at the expense of making them "fatter". The number of layers you can use will depend upon the wire size and available total diameter of the coil.

You can see that there are many interrelated factors for a practical design.

 

[ft92]

Thank you for your answer, but actually I'm only asked to find the number of turns without researching the properties of the material.

 

[gneill]

So it looks like you've found a viable result. Just make sure that answer is given to an appropriate number of significant figures.

 

[TSny]

Maybe relevant: https://van.physics.illinois.edu/qa/listing.php?id=2353

 

[gneill]

Maybe relevant: https://van.physics.illinois.edu/qa/listing.php?id=2353

Indeed. I hadn't thought about the fact that the fringe effect would significantly affect the field strength at the ends of a finite solenoid. As a good approximation, assume it cuts the calculated field value in half. Or, are you expected to derive an equation for the field strength of a finite solenoid?