My Friends,
Electromagnetic Induction is at the very root of Todays Technological Society! Its what makes our Wheels go around!
Today I want to give an Example, and show another inconsistency in the Equation:
E.M.F = -N δΦB / δt
First, lets give some background! German physicist Franz Neumann, produced a quantitative statement of Faraday's second law of electromagnetic induction:
The magnitude of an electromagnetically induced e.m.f. (E) is given by E = −dΦ/dt, where Φ is the magnetic flux. This is a quantitative statement of Faraday's second law of electromagnetic induction and is sometimes known as the Faraday–Neumann law. This law was stated quantitatively by the German physicist Franz Neumann (1798–1895) in the 1840s.
Ref: Oxford Reference
So, what we use today, is really, the: Faraday–Neumann Law.
Example:
First, the Symbol Delta: Lowercase Symbol ( δ ) as well as Uppercase Symbol ( Δ ) or just d:
Delta Symbol: Change
Uppercase delta ( Δ ) at most times means “change” or “the change” in maths. Consider an example, in which a variable x stands for the movement of an object. So, “Δx” means “the change in movement.” Scientists make use of this mathematical meaning of delta in various branches of science.
The Faraday–Neumann Law Equation is: E.M.F = -N δΦB / δt
Where:
- E.M.F is measured in Units of Volts.
- The - Negative Sign represents Lenz's Law, the Sine Opposite of the Source.
- N is the Turns of the Inductor that has a Voltage Induced.
- δΦB is the Change in Magnetic Flux B, from BMax to BMin.
- δt is the Change in Time, in Seconds - This is important, focus on this for the most part.
We all know, the Reciprocal of Frequency is: 1 / f = t in Seconds. This is true the other way around also: 1 / t = f in Hertz. Lets take a simple example:
The Frequency Reciprocal: 1 / 50 Hz = 0.02 Seconds, and we can also verify this: 1 / 0.02 Seconds = 50Hz. To simplify: 1 / 1 Hz = 1 Second, simple right!
Here we must observe, 1 Cycle, or 1 Period has the following Sinusoidal Waveform:
The same is true for the following Single Cycle Alternating Square Waveform:
Hold this thought for a minute, because here we see a situation where we have a Logical Error.
First lets run through the rest of the Math. We are going to use the following figures:
- N = 100.
- δΦB = 100.05 Gauss - 0.5 Gauss = 100 Gauss. Remember, δ is the Change in the Variable.
- δt = 1 Second - 0 Seconds = 1 Second. Remember, δ is the Change in the Variable.
We are going to use a Cross Sectional Area of:
- CSA = 100 cm's x 100 cm's = 1.0 Sq meters
Lets Calculate:
- Min = 100.0 Turns x CSA x ( 0.5 Gauss / 10000 Gauss in One Tesla ) x Cos( Theta θ = 0.0 ) = 0.005
- Max = 100.0 Turns x CSA x ( 100.0 Gauss / 10000 Gauss in One Tesla ) x Cos( Theta θ = 0.0 ) = 1.0
Where:
- Theta θ is the Angle of the Magnetic Flux, where θ = 0.0 means Magnetic Flux is Perpendicular to the Axis of the Inductor.
Now we can calculate the E.M.F: - (1.0 - 0.005) / 1.0 Second = -0.995 Volts
The Logical Error:
During a Single Cycle, we have two Peaks of BMax, and also three instances of BZero, this should be an issue:
Logically, we should take the Minimum Value of B, and the Maximum Value of B and take the Difference, in accordance with the Definition of the Delta Symbol! So one would think Logically, that from Zero Degrees to 90 Degrees is the Minimum to Maximum Value Flux B that we could measure, this would be the Delta of B required to "Generate" a an E.M.F measured in units of Volts, for a given Change in Flux ΦB.
However, the Math does correctly Calculate the Induced Voltage! In saying this, we are not using the true : δΦB = BMax - BMin = δΦB! This is misleading and shows an Inconsistency in the Logic of the very Delta Definition!
It should be obvious that a Gentleman's Agreement has been made to over look this Inconsistency and not worry about Correctness anymore!
This shows another example, where Waveform and Wave Type is important!
The intensive study of these Fundamental Laws continues to show Inconsistencies! Sadly!
Best Wishes,
Chris
