How come the length of a wire does not affects on the circuit current?
Today we started learning about the electromagnetic induction. Out teacher gave us the following explanation:
Suppose we have a conduction frame inside a magnetic field $\vec{B}$ going towards the page. That frame has a stationary edge (also a conductor), which we will call $AB$. $AB$ is pushed right with a starting velocity $\vec{v}$. Lets say $AB$ moves at a constant speed. As a result of the edge's movement, Lorentz force acts on it's inside charges. It's direction on the positive charges is up (towrads $A$), and on the negative charges, down (towards $B$). 395
This seperation creates an electric potencial $V_{AB}$, which will cause an electric force $\vec{F_E}$ to act on the edge's charges as well. The situation will balance when:
$$F_B = F_E$$.
Thus
$$qvB=qE$$
We know that $E=\frac{V_{AB}}{d}$, where $d$ is the length of $AB$. Thus:
$$qvB=q\frac{V_{AB}}{d}$$
Thus
$$V_{AB}=vBd$$
This represents the voltage created between the two sides of the edge, which moves inside a magnetic filed. It will, obviously, create a current in the circuit, as follow:
$$I=\frac{V_{AB}}{R}=\frac{vBd}{R}\quad (1)$$
Where $R$ is the resistance of the circuit.
From the last equation, we can learn that a longer edge $d$, will produce a larger current $I$. So far so good.
But, if we dig down in the definition of the resistance of a wire, we will remember that:
$$R = \rho\frac{d}{A} \quad (2)$$
Where $d$ is the length of the conductor, $A$ is the cross-sectional area of the conductor and $\rho$ is the electrical resistivity. The last formula states that there is a direct relation between $R$ and $d$. If we combine $(1)$ and $(2)$, we produce the follow:
$$I=\frac{vBd}{\rho\frac{d}{A}}=\frac{vBA}{\rho}$$
Wait - the last formula shows us that the current $I$ has no relation to the length of the wire $d$ - but how is that possible? Moving a million metre wire and a one meter wire inside a magnetic field will produce the same current? My teacher said that isn't true, but we had no time to find the error, if such one trully exist.
Thanks in advance :)
P.S - i'm just a highschool student, so please use math as simple as possible to answer. Thanks. 780