Electromagnetic Induction: Diagram-Based Questions (7)

Question

A rod of length $L = 0.5\,\text{m}$ slides along frictionless rails with velocity $v = 4\,\text{m/s}$ in a uniform magnetic field $B = 0.2\,\text{T}$ pointing out of the page. The circuit resistance is $R = 2\,\Omega$. Find the induced EMF and the current. The diagram shows the rod sliding right.

Solution — Step by Step

The induced EMF is $0.4\,\text{V}$ and current is $0.2\,\text{A}$ flowing clockwise.

Why This Works

Motional EMF comes from the magnetic force on the free electrons in the moving rod. The force $qv \times B$ pushes electrons to one end, creating a potential difference. The formula $\varepsilon = BLv$ packages this neatly when $v$, $B$, and $L$ are mutually perpendicular.

Lenz’s law gives the direction: the induced current always opposes the change in flux. If flux is increasing out of the page, the induced current creates a field into the page inside the loop — meaning clockwise circulation.

Alternative Method

Use Faraday’s law directly. The flux is $\Phi = BLx$ where $x$ is the rod’s position.

$\varepsilon = -\frac{d\Phi}{dt} = -BL\frac{dx}{dt} = -BLv$

The magnitude is $0.4\,\text{V}$. The minus sign tells us the direction opposes the increase. Same physics, same answer.

Common Mistake

Mixing up the direction of the induced current. The fix: always state whether flux is increasing or decreasing, then apply Lenz’s law. JEE Main 2023 had a question where the field was into the page — a single sign flip changed the answer.