Question

(a) A rod of length l is moved horizontally with a uniform velocity 'v' in a direction perpendicular to its length through a region in which a uniform magnetic field is acting vertically downward. Derive the expression for the emf induced across the ends of the rod.

(b) How does one understand this emotional emotion by invoking the Lorentz force acting on the free charge carriers of the conductor? Explain.

Answer 1

Here we have to discuss induced EMF which is induced due to the relative motion between the magnet and current conducting rod hence we have to relate or understand this law in reference to magnetic field.

Complete step by step answer:
(a) As we all know that the change in magnetic flux $\dfrac{{d\phi }}{{dt}}$ is given by:
$\dfrac{{d\phi }}{{dt}} = Bvl$
Here, B is the magnetic field, v is the velocity of the conductor and l is the length of the rod.
We know that the change in magnetic flux is equal to induced emf $\varepsilon$.
Therefore, we can say that the induced emf is:
$\varepsilon = Bvl$
(b) As the conductor is in equilibrium then, we can say that the Lorentz force F acting on the conductor is zero.
$F = qE + qvB\sin \theta$
Here, E is the electric field, q is the charge, v is the velocity and $\theta$ is the angle between velocity and Magnetic field which is equal to $90^\circ$ as given in the question.
Hence, we can say that:
$\begin{array}{l} \Rightarrow 0 = qE + qvB\sin 90^\circ \\\ \Rightarrow E = - Bv\\\ \Rightarrow \left| E \right| = Bv \end{array}$
Since we also know that $\dfrac{{d\varepsilon }}{{dl}} = \left| E \right|$ and hence we can write the above equation as:
$\begin{array}{c} \dfrac{{d\varepsilon }}{{dl}} = Bv\\\ \int {d\varepsilon } = \int {Bvdl} \\\ \varepsilon = Bvl \end{array}$
Hence the induced emf by analysing Lorentz force is also equal to Bvl.

Additional Information: According to Lenz law, the magnetic field created by induced current opposes the initial changing magnetic field which produced it. The direction of this current due to the initial changing magnetic field is always given by Fleming’s right hand rule. According to Lenz law, if a magnet is to be a move towards a current-carrying coil or vice-versa then the magnetic field produced works in such a way that it always opposes the motion of the current-carrying coil

Note: Finally, we can conclude that emf can be induced by changing magnetic flux linking with the coil. Some important applications of induced emf are in generators, galvanometers and transformers. More is the rate of change of magnetic flux, more is the magnitude of induced emf.