Maxwell- Faraday's Equation, MotionalEmf, Work Done by Lorentz Force

A moving rod placed in a stationary U-shaped frame.

When a conductor (the moving rod in this case) moves perpendicular to a magnetic field, an electromotive force (EMF) is induced according to Faraday's law of electromagnetic induction. The rate of change of magnetic flux through the loop formed by the moving rod and the stationary U-shaped rods is given by = B⋅L⋅V where B is the magnetic field strength,L is the length of the rod, and v is its velocity.

Now, the negative of rate of change of magnetic flux = work done by electromagnetic force throughout the loop, which is BLV+IR, i=current , r=resistance

But this leads to that,BLV=BLV+I--->>IR=0 , means electric current through loop is 0, So, Just Please Clarify My Confusion

The emf induced in a rod of length $$\ell$$ travelling with speed $$v$$ in a plane perpendicular to a magnetic field $$B$$ is $$B\ell v$$.
If the resistance of the complete circuit is $$R$$ then the power dissipated in the circuit is $$\dfrac{(B\ell v)^2}{R}= \dfrac{B^2\ell^2 v^2}{R}$$.
A current carrying conductor moving in a magnetic field experiences a force acting on it $$B I\ell$$ where $$I$$ is the current in the circuit, $$\dfrac{B\ell v}{R}$$.
For the rod to move at constant velocity, $$v$$, an external force, $$\dfrac{B\ell v}{R}$$ must be applied to the rod and the work done per second by the external force is $$B I\ell\cdot v = B\cdot \dfrac{B\ell v}{R}\cdot \ell\cdot v = \dfrac{B^2\ell^2 v^2}{R}$$ which is exactly equal to the power dissipated in the circuit.
• The statement the negative of rate of change of magnetic flux = work done by electromagnetic force throughout the loop, which is BLV+IR is not correct. $B\ell v$ is the induced emf and $IR$ is the potential difference across the resistance $R$. These two quantities are equal in magnitude which means that the current in the circuit is $B\ell v/R$ Nov 1, 2023 at 8:30
• @MultiversalExplorers - You either treat the $B\ell v$ as an emf on the right hand side of the equation or the $B\ell v$ as a potential difference on the left hand side of the equation. That term cannot be on both sides of the equation. Nov 2, 2023 at 8:26