P0986 Electromagnetism › Electromagnetic Induction

Induced Current in a Split Metal Frame

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Electromagnetic Induction Beginner Electromagnetic induction

Source: High school physics (Chinese)

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Electromagnetic induction

Problem

$CDEF$ is a metal frame lying in a uniform magnetic field directed into the page. A conducting rod $AB$ slides along the frame, dividing it into two loops: $ABEF$ on the left and $ABCD$ on the right, each containing a galvanometer. The rod $AB$ is moved to the right with velocity $v$.
  1. Determine the direction of the induced current in circuit $ABCD$.
  2. Determine the direction of the induced current in circuit $ABEF$.
Problem image

Circuit $ABCD$: $A \to B \to C \to D \to A$ (clockwise). Circuit $ABEF$: $A \to B \to F \to E \to A$ (counterclockwise).

In the moving rod the magnetic force on a positive charge is $q\vec{v}\times\vec{B}$. With $\vec{v}$ to the right and $\vec{B}$ into the page, this force points from $A$ to $B$ (upward), so $B$ is the high-potential end and the rod acts as the source, driving current from $A$ to $B$ inside it.

Circuit $ABCD$ (right): as the rod moves right this loop shrinks, so the into-page flux through it decreases. By Lenz's law the induced current is clockwise to maintain the flux: $A \to B \to C \to D \to A$.

Circuit $ABEF$ (left): as the rod moves right this loop grows, so the into-page flux increases. By Lenz's law the induced current is counterclockwise to oppose it: $A \to B \to F \to E \to A$.

Both results agree that the current in the shared rod flows from $A$ to $B$.