- #1
FranzDiCoccio
- 342
- 41
Hi all,
Kirchhoff's equation for this simple circuit is equivalent to
[tex]\dot I=\frac{V}{L}[/tex]
Where [itex]V=V_0 \sin(\omega t)[/itex]. Integrating both sides should give
[tex]
I(t) = -\frac{V_0}{L\omega} \cos(\omega t)+c
[/tex]
where [itex]c[/itex] is an arbitrary constant (current).
Here, most of the derivations I've found simply drop the [itex]c[/itex], without so much as a word. Actually they would not even mention it in the first place.
Of course keeping that constant does not seem physically right. What could the value of [itex]c[/itex] possibly be? How would one determine it?
It should be ruled out by a boundary condition, but I am not really able to put my finger on it.
Somewhere I read: the current intensity must be maximum when the emf is zero... but that seems kind of a hindsight.
The only things I can think of are based on symmetry considerations. The "driving" potential is symmetric, so I expect the current to be symmetric too.
There is only one aspect that I can think of that could break the symmetry and introduce a "background" constant (DC) current. That would be the rotation direction of the coil in the AC generator.
But this makes me think that if the rotation of the coil is inverted, so would be the value of [itex]c[/itex].
So if the coil spins clockwise we get e.g. [itex]c[/itex], and if it spins counterclockwise we get [itex]-c[/itex].
On the other hand, the emfs of these two cases look exactly the same (especially if I do not know when the spinning has started). They only differ by a phase.
This makes me think that they should generate the same alternating current, which means [itex]c=-c[/itex], which of course means [itex]c=0[/itex].
Am I making any sense?
Kirchhoff's equation for this simple circuit is equivalent to
[tex]\dot I=\frac{V}{L}[/tex]
Where [itex]V=V_0 \sin(\omega t)[/itex]. Integrating both sides should give
[tex]
I(t) = -\frac{V_0}{L\omega} \cos(\omega t)+c
[/tex]
where [itex]c[/itex] is an arbitrary constant (current).
Here, most of the derivations I've found simply drop the [itex]c[/itex], without so much as a word. Actually they would not even mention it in the first place.
Of course keeping that constant does not seem physically right. What could the value of [itex]c[/itex] possibly be? How would one determine it?
It should be ruled out by a boundary condition, but I am not really able to put my finger on it.
Somewhere I read: the current intensity must be maximum when the emf is zero... but that seems kind of a hindsight.
The only things I can think of are based on symmetry considerations. The "driving" potential is symmetric, so I expect the current to be symmetric too.
There is only one aspect that I can think of that could break the symmetry and introduce a "background" constant (DC) current. That would be the rotation direction of the coil in the AC generator.
But this makes me think that if the rotation of the coil is inverted, so would be the value of [itex]c[/itex].
So if the coil spins clockwise we get e.g. [itex]c[/itex], and if it spins counterclockwise we get [itex]-c[/itex].
On the other hand, the emfs of these two cases look exactly the same (especially if I do not know when the spinning has started). They only differ by a phase.
This makes me think that they should generate the same alternating current, which means [itex]c=-c[/itex], which of course means [itex]c=0[/itex].
Am I making any sense?