Saturday, August 13, 2022

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

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 


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