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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