### Related Formula The motional EMF induced across a moving conductor of instantaneous length ell$\ell$ inside a perpendicular uniform magnetic field is given by: E = B cdot ell cdot v$E = B \cdot \ell \cdot v$ ### Core Logic Let the V-shaped guide rails form an \angle, so that the instantaneous length ell$\ell$ of the conducting \bar grows linearly with its horizontal position distance x$x$ from the vertex [cite: 782, 791]: ell propto x$\ell \propto x$ Since the \bar moves with a constant velocity v$v$, its displacement position at any time t$t$ is : x = v cdot t implies ell propto v cdot t$x = v \cdot t \implies \ell \propto v \cdot t$ Substituting this time-dependent length into the induced EMF expression : E = B cdot ell cdot v implies E propto B cdot (v cdot t) cdot v implies E propto t^1$E = B \cdot \ell \cdot v \implies E \propto B \cdot (v \cdot t) \cdot v \implies E \propto t^1$ Comparing this to the given relation E propto t^n$E \propto t^n$ gives the exponent[cite: 188, 791]: n = 1$n = 1$ ### Step 1: Geometric Analysis The expanding circuit loop configuration across time is shown below:

### Pattern Recognition For \parallel rails, the length ell$\ell$ remains constant, meaning induced EMF is independent of time (E propto t^0$E \propto t^0$). For V-shaped divergent rails, the effective length increases linearly with distance, making the induced EMF directly proportional to time (E propto t^1$E \propto t^1$). ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Physics: Electromagnetic Induction