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Your finding of $I\propto L$ from the Lorentz law is based on the premise that, mathematically, if $A\propto B$ and $A\propto C$ it necessarily follows that $B\propto C$. But that is only true if $B$ and $C$ are independent. Here, $B$ and $C$ are current and conductor length, which are not independent.

The root of the problem is you are attempting to compare the current in Ohm's law and the current in Lorentz's law. This is like, as they say, "comparing apples to oranges". Ohm's law deals with the electric force that drives current through a conductor with resistance due to an electric field that is parallel to the current. The Lorentz law for a current carrying conductor describes the magnetic force that acts on a conductor as a result of a given amount of current (which is determined by Ohm's law) perpendicular to the magnetic field.

Bottom line: There is no contradiction. Current and conductor length are not independent variables and the two laws involve different forces with different roles.

Hope this helps.

Your finding of $I\propto L$ from the Lorentz law is based on the premise that, mathematically, if $A\propto B$ and $A\propto C$ it necessarily follows that $B\propto C$. But that is only true if $B$ and $C$ are independent. Here, $B$ and $C$ are current and conductor length, which are not independent.

The root of the problem is you are attempting to compare the current in Ohm's law and the current in Lorentz's law. This is like, as they say, "comparing apples to oranges". 

Ohm's law deals with the electric force that drives current through a conductor with resistance due to an electric field that is parallel to the current. 

The Lorentz law for a current carrying conductor describes the magnetic force that acts on a conductor as a result of a given amount of current (which is determined by Ohm's law) perpendicular to the magnetic field.

Bottom line: There is no contradiction. Current and conductor length are not independent variables and the two laws involve different forces with different roles.

Hope this helps.

Your finding of $I\propto L$ from the Lorentz law is based on the premise that, mathematically, if $A\propto B$ and $A\propto C$ it necessarily follows that $B\propto C$. But that is only true if $B$ and $C$ are independent. Here, $B$ and $C$ are current and conductor length, which are not independent.

The root of the problem is you are attempting to compare the current in Ohm's law and the current in Lorentz's law. This is like, as they say, "comparing apples to oranges". Ohm's law deals with the electric force that drives current through a conductor with resistance due to an electric field that is parallel to the current. The Lorentz law for a current carrying conductor describes the magnetic force that acts on a conductor for a given the amount of current (which is determined by Ohm's law) perpendicular to the magnetic field.

Bottom line: There is no contradiction. Current and conductor length are not independent variables and the two laws involve different forces with different roles.

Hope this helps.

Your finding of $I\propto L$ from the Lorentz law is based on the premise that, mathematically, if $A\propto B$ and $A\propto C$ it necessarily follows that $B\propto C$. But that is only true if $B$ and $C$ are independent. Here, $B$ and $C$ are current and conductor length, which are not independent.

The root of the problem is you are attempting to compare the current in Ohm's law and the current in Lorentz's law. This is like, as they say, "comparing apples to oranges". Ohm's law deals with the electric force that drives current through a conductor with resistance due to an electric field that is parallel to the current. The Lorentz law for a current carrying conductor describes the magnetic force that acts on a conductor as a result of a given amount of current (which is determined by Ohm's law) perpendicular to the magnetic field.

Bottom line: There is no contradiction. Current and conductor length are not independent variables and the two laws involve different forces with different roles.

Hope this helps.

If I understand you correctly, you are saying there seems to be a conflict between Ohms law, where $I\propto 1/L$, and the Lorentz force law where $I\propto L$ follows from $F\propto L$ and $F\propto I$. If that’s the case then you are, as they say, "comparing apples and oranges".

The current in a conductor in a magnetic field is not due to the Lorentz force, it’s the other way around. The current in the conductor is due to the force of the electric field in the conductor, not the external magnetic field.

So there is no conflict since the current from Ohm’s law is due to a different force.

Hope this helps.

Your finding of $I\propto L$ from the Lorentz law is based on the premise that, mathematically, if $A\propto B$ and $A\propto C$ it necessarily follows that $B\propto C$. But that is only true if $B$ and $C$ are independent. Here, $B$ and $C$ are current and conductor length, which are not independent.

The root of the problem is you are attempting to compare the current in Ohm's law and the current in Lorentz's law. This is like, as they say, "comparing apples to oranges". Ohm's law deals with the electric force that drives current through a conductor with resistance due to an electric field that is parallel to the current. The Lorentz law for a current carrying conductor describes the magnetic force that acts on a conductor for a given the amount of current (which is determined by Ohm's law) perpendicular to the magnetic field.

Bottom line: There is no contradiction. Current and conductor length are not independent variables and the two laws involve different forces with different roles.

Hope this helps.