**Plus42 and Casio fx-9750GIII: Magnetic Flux Density of a Solenoid**

**Introduction: Ampère's Circuital Law**

A solenoid is an electromagnetic coil or wire that controls a magnetic field. The solenoid is a key part of the automobile starter.

Constant Radius: If the solenoid has a constant radius, the simplest case, the magnetic flux density is calculated as:

B = μ0 * N * I / L

where

μ0 = permeability in a vacuum. More about this later.

N = number of turns

I = current in Amps

B = magnetic flux density in Telsa (T, or Wb/m^2, or kg/(A*s^2)

Various Radii: If the radii vary in the solenoid, we can calculate the sum of the magnetic flux density per each point z_i and radius R_i.

B_i = μ0 * I * N * R_i^2 / (2 * (R_i^2 + z_i^2)^1.5)

**The μ0 Constant: Or Is It?**

The μ0 represents the permeability in a vacuum, and traditionally, it is a constant defined as:

μ0 = 4 * π * 10^-7 T * m/A ≈ 1.25663706144 * 10^-6 T * m/A

However, CODATA has a slightly different value assigned, here is the May 2019 value:

μ0 ≈ 1.25663706212 * 10^-6 T * m/A

Since the traditional value is reasonably close, I am going to use it the programs presented here.

**Casio fx-9750GIII Program: SOLENOID**

Option 1: Constant Radius

Option 2: Various Radii, the total flux is calculated

Program:

4×π×10^(-7)→U

Menu "SOLENOID","CONSTANT RADIUS",1,"DIFFERENT RADII",2

Lbl 1

"TURNS"?→N

"CURRENT (A)"?→I

"LENGTH (M)"?→L

Goto 3

Lbl 2

"TURNS"?→N

"CURRENT (A)"?→I

0→B

"NO. OF RADII"?→J

For 1→K To J

"DATA"

K⊿

"RADIUS"?→R

"LENGTH"?→Z

B+(U×I×N×R)÷(2×(R^2+Z^2)^1.5)→B

Next

Goto 3

Lbl 3

"FLUX DENSITY:"⊿

B

**Plus42 (HP 27S/19B/17B) Equations: SOLENOID**

Constant Radius:

SOLE1:B=4×π×ALOG(-7)×N×I÷L

Various radii (single calculation):

SOLE2:B=(4×π×ALOG(-7)×I×N×R)÷(2×(R^2+Z^2)^1.5)

Use a register to store and sum B_i.

**Examples **

Constant Radius:

N = 8000 turns

L = 0.06 m

I = 30 A

B ≈ 5.026548246 Telsa

Various radii (total):

N = 2500 turns

I = 15A

(point z, radii r)

z_1 = 0 m, r_1 = 0.6 m

z_2 = 0.5 m, r_2 = 1 m

z_3 = 0.8 m, r_3 = 0.7 m

B ≈ 0.09604006539 Telsa

Sources

"CODATA Recommended Values of the Fundamental Physical Constants: 2018" National Institute of Standards and Technology. United States Department of Commerce. May 2019 https://physics.nist.gov/cuu/pdf/wall_2018.pdf Retrieved June 7, 2022.

"Solenoid" Wikipedia. Last edited April 30, 2022. https://en.wikipedia.org/wiki/Solenoid Accessed June 1, 2022.

Nave, Carl Rod. "Solenoid as Magnetic Field Sources" HyperPhysics Georgia State University. 2016 http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/solenoid.html Accessed June 3, 2022.

This blog entry is dedicated to Chris B.

Eddie

All original content copyright, © 2011-2022. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.