Final answer:
The induced emf across the ends of a rod moving through a magnetic field is calculated using Faraday's law of induction, with the magnitude given by ε = Blv, where B is the magnetic field, l is the length of the rod, and v is the velocity of the rod.
Step-by-step explanation:
To derive the expression for the electromotive force (emf) induced across the ends of a rod moving through a magnetic field, we use Faraday’s law of induction. When a rod of length l is moved horizontally with uniform velocity v perpendicular to its length in a region with a uniform magnetic field B acting vertically downward, the magnetic flux through the area swept by the rod changes. This change in flux induces an emf.
The area swept by the rod in time t is l times the distance moved vt, and hence the change in magnetic flux is given by ΔΦ = B(lvt). According to Faraday's law, the induced emf (ε) is equal to the negative change in flux over time, thus ε = -dΦ/dt.
By differentiating the flux Φ = Blvt with respect to time t, we obtain ε = -B(lv), since B and l are constants. The negative sign indicates the direction of the induced emf, as per Lenz's Law. Therefore, the magnitude of the induced emf is ε = Blv.