...sing along here!

Magnetic effects which attract or repel
Are detected in what's called a field,
And despite being hidden from our human eyes
Then these forces, as 'lines', are revealed.
These lines always travel from North towards South,
Making field strength a vector, you see,
And the closer together the greater the strength
Or 'magnetic flux density', B.

As B involves area, sometimes we'd rather
Use B = φ per square metre
So φ is magnetic flux, measured in Webers,
And φ = AB cos θ.
But flux here is only for one turn. Supposing
A coil of N turns should draw nigh?
In this case we'd have to make use of 'flux linkage'
Where total flux = N φ.

...sing along here!

Spinning electrons make both current and field,
Always together, as neither can escape
Because the field's direction relies on current flow
And each conductor has its own field shape.
REFRAIN:
We use the Right Hand Grip Rule to save the day,
We point our thumb where the current's flowing
And then our curving fingers will show the way
Round which the lines of force are going.

Long straight conductors have concentric lines of force
around them, But the question is, which way?
We use the Right Hand Grip Rule, and for its strength
B = μ₀ I by 2 π a.
REFRAIN
Flat coils of circles have a different field,
The Right Hand Grip Rule makes directions clear,
And at the circle's centre the field strength B
Equals μ₀ I by 2 times r.
REFRAIN
Finding the north pole of a solenoid
Is easy if you use the Grip Rule well.
Along its axis B is μ₀ I times 'turns by length';
B = μ₀ I 'big N' by l...
(into BRIDGE)
But if we have a solenoid with soft iron core,
Permeability's a different sort
Because the metal's there we need to use μ_r ,
Which is the iron's μ by μ₀... and then...
REFRAIN

...sing along here!

REFRAIN:
For fields and forces Fleming's Left Hand Rule is your selection;
The first finger points out the field, and points in its direction,
The second finger shows the current at the intersection*,
The thumb's the force, and motion of the wire's deflection.

You put a wire with current in a field, it so won't stay,
As lines of force won't cross, the fields arrange a different way.
The length of wire that's normal in the field is l, agreed -
So F is B I l is the formula you need.
REFRAIN
Now current has a flow of charge, we know the statement's true,
If current causes force to act, then charges will do too.
A charge of Q that's travelling with v velocity
Will require a calculation from F = B Q v.
REFRAIN
Electrons travel backwards when compared to current flow,
So the second finger's opposite to where electrons go.
So negative charged particles when travelling, like e,
Normal to a field will make a force F = B e v.
REFRAIN

...sing along here!

Two conductors carry currents, 'a' metres apart,
But each has a field, and so the wires will interact,
So the wires exert a force, and when the currents flow
In the same direction in each wire, they will attract.
REFRAIN:
Two conductors in a vacuum, one metre apart,
Which are straight, thin, parallel and infinitely long
Have an amp of current flowing when they both exert a force
Of μ₀ by 2 Newtons on each metre of each one.

The product of the currents flowing in each wire
By 2 π a, and multiplied by μ₀ L
Gives the size of such a force, and when the currents flow
In opposite directions in each wire, they will repel.
REFRAIN

...sing along here!

Emf is induced in conductors when magnetic flux changes in a field;
By combining two laws of inductors a connecting equation is revealed.
Faraday's Law states that emf is proportional to rate of change of φ,
Lenz's Law says that the current will flow
To oppose the change that made it, it will try:
REFRAIN:
E equals minus d N φ, all by d t
Will help you to derive induction formulae...

For a wire cutting magnetic flux lines at right angles then φ = AB,
It sweeps area A in a second, so we now replace A with L times v,
Because it does this in the magical time of a single second our work is done,
E = minus B L v... it's simple now because we made d t equal one...
REFRAIN
For a coil rotating in a magnetic field at any time, φ is cos θ times AB,
But we know that θ equals ω t, where the time since θ's zero is t.
E equals minus B A N cos ω t by d t, where N is number of turns,
Differentiate so E is B A N ω sin ω t ... it's simple now 'cause one learns... that...
REFRAIN
(into BRIDGE)
A search coil measures the magnetic flux density of magnetic fields which do change;
Because the flux changes, emf is induced: there are lots of equations to arrange.
Consider the case when the field becomes zero, so the flux will as well, in time t,
Emf equals minus d N φ by d t when the original flux is N A B.
But R's the resistance of the circuit, and Ohm's Law states that emf will equal R I,
So I R will equal B A N over t, where B A N was the total change in φ.
And Q is the charge which flows in the coil, and we know that I t equals Q,
So we're left with Q equals B A N over R: It's this equation made it easy for you...
REFRAIN

...sing along here!

If the current changes in a coil then the field will change around the coil
And an emf's induced in that conductor,
And because back emf occurs in the same coil as the current flows
Then we call that special coil a self-inductor.
REFRAIN:
Self-inductance is defined, if L's the self-inductance,
As the emf's L d I by d t:
If V's one volt when current changes at one amp per second
Then the self-inductance of the coil's one Henry.

There's some metal moving in a field but it's not a lovely steady field
And because it's moving, it's as if it's changing,
So an emf's induced inside, making currents flow around inside
Known as eddy currents with effects wide-ranging.
REFRAIN
Eddy currents can be rather large as they follow low-resistance paths
As we know from V equals I R, Ohm's Law,
They can have magnetic values, like damping oscillating meters, or
They can heat the metal, to bake a ceramic core.
REFRAIN

...sing along here!

Resistors in a circuit will resist the current flowing
(As long as the potential of the circuit isn't growing!)
Ohm's Law states that current is proportional to V:
They're both in phase and varying with sine ω t.
REFRAIN:
Alternating current series circuits, Alternating current series circuits...

The potential which inductors have to start with is so large
It induces emf opposing rate of flow of charge.
When V decreases, I can then increase, and so you see
V₀ cos ω t and I₀ sin ω t.
REFRAIN
Current in capacitors flows when they are connected,
Charge builds up upon their plates as quickly as it is injected.
But more charge means V increases: I must dwindle rapidly -
I₀ cos ω t and V₀ sin ω t.
REFRAIN

...sing along here!

It's hard to take an average of ac straight away
For an alternating current varies sinusoidally,
So we use a 'root mean square' instead, or 'RMS' we say
Though it should be 'square mean root' as that's the order of the three.
REFRAIN:
And the RMS value of ac Is the same as the steady value of dc
Which will dissipate, it will dissipate The circuit's energy at the same rate.

So you square the peak and lowest values now without delay,
For to take a mean, positive must all the values be,
And then you take the mean for one whole cycle of the wave
Which must have a square root taken to be RMS ac.
REFRAIN
In resistors V and I both change with sine waves, yes they do,
So the minimum is zero and the peak is 1, you see.
Square and mean: we find the average is one over two,
Then we root: RMS - peak by root two for I or V.

Find the average power dissipated in resistors when we know that Power = I V.
I's I₀ sin ω t, V's V₀ sin ω t, so Power's I₀ V₀ sin² ω t
As the average value of sin² ω t's a half we found when working out the RMS,
So the average Power's I₀ V₀ over 2, which should answer all your questions, more or less.

Find the average power dissipated in inductors when we know that Power = I V.
I's I₀ sin ω t, V's V₀ cos ω t; Power multiplies together as we see.
But the average effect of sin and cos is zero, and this makes our mathematic powers reach their prime
As the average Power dissipated in inductors will be zero, zero, zero every time.

Find the average power dissipated in capacitors; we know that Power = I V.
I's I₀ cos ω t, V's V₀ sin ω t; Power multiplies together as we see.
But the average effect of sin and cos is zero, and this makes our mathematic powers reach their prime
As the average Power dissipated in capacitors is be zero, zero, zero every time.

You've a mission to derive a reactance expression,
A capacitative one, X_C.
And the mission's aim lies also in the answer to this question
Which is one over ω C.
REFRAIN:
In capacitors the current leads potential difference V,
And by ninety degrees every time,
So current is peak current times by cos ω t,
And the voltage must vary with sine.

The capacitative current is the rate of flow of charge,
So I is d Q by d t.
A capacitor's potential is proportional to charge,
So d Q equals d (C V).
REFRAIN
If you put them both together, d (C V) by d t
And substitute in then for V,
I is d C V₀ sin ω t, all by d t:
Differentiate now, happily.
REFRAIN
Differentiating sin ω t is not a chore
And can be done most easily
If you change the sin to cos, and add an ω before;
C V₀ ω cos ω t.
REFRAIN
And we know that I₀ cos ω t is also I,
So I₀ is V₀ ω C
And we know Ohm's Law where X_C equals V over I;
We have one over ω C!
REFRAIN
You've a mission to derive a reactance expression,
A capacitative one, X_C.
And the mission's aim lies also in the answer to this question
Which is one over ω C.

We have an inductor all shiny and new.
Its back emf is proportional to
The rate of change of current gone through.
Derive the reactance, X_L.
REFRAIN:
Inductors in ac circuitry, where V leads I or I lags V,
So V is V₀ cos ω t,
And current will follow, a sine.

The inductance equation we now will fragment.
We know the rate of change of current
Is a.k.a. d I by d t,
And back emf is called V.
REFRAIN
If I is I₀ sin ω t,
By calculus find an expression for V.
If V follows cos and we substitute, well
V is I₀ ω L.
ALTERED REFRAIN:
Inductors in ac circuitry, where Vs lead Is or Is lag Vs,
The gap between is ninety degrees,
We're reaching the very last line!

With V, with I, we've also Ohm's Law
V = I R - you've done this before.
We just rearrange, at which we 'excel':
Reactance is ω L!
ALTERED REFRAIN:
Inductors in ac circuitry, where V leads I or I lags V,
So V is V₀ cos ω t,
Reactance is ω L!

...sing along here!

REFRAIN:
Resonance, resonance, when impedance in a circuit is a minimum,
Resonance, resonance, then the current in a circuit is a maximum.

In series circuits of ac can be R, X_L and X_C,
The reactance and resistance in a circuit has a vector sum:
To calculate this without fuss, we can use good ol' Pythagoras,
And we have 'impedance' of the circuit, with no problem...
REFRAIN
Impedance has an all-time low when it's equal to resistance alone,
The reactances are equal so X_L is the same as X_C,
With such little opposition of charge the resulting circuit current is large,
This is said to happen at a 'resonant frequency'...
REFRAIN
We derived opposition of flow, or 'reactances' a while ago:
X_L's ω L, and X_C's one over ω C;
As we know at f₀ these are the same, rearranging is the name of the game,
ω is 2 π times the resonant frequency...
REFRAIN
(into BRIDGE)
2 π L f₀ is equal to one over 2 π C f₀
So f₀ is one over 2 π root (L C), woah...
2 π L f₀ is equal to one over 2 π C f₀
So f₀ is one over 2 π root (L C),
What we have is...
REFRAIN