Right-Hand Rules - Physics 2

1. Which end of the rod becomes positively charged due to the magnetic force on the free electrons in the rod, and what is the direction of the induced electric field inside the rod once electrostatic equilibrium is reached?
   Answer: The end at $(L, 0, 0)$ becomes positive, and the induced electric field inside the rod points in the $-\hat{x}$ direction (from the $(L,0,0)$ end toward the origin, i.e., from positive to negative), opposing the continued magnetic force on the electrons and establishing equilibrium. When a conducting rod moves through a magnetic field, the free electrons inside experience a magnetic force given by $\vec{F} = q\vec{v} \times \vec{B}$. Since electrons carry negative charge, their force is opposite to $\vec{v} \times \vec{B}$. Start by computing the cross product: $\vec{v} \times \vec{B} = v_0\hat{y} \times B_0\hat{z} = v_0 B_0(\hat{y} \times \hat{z}) = v_0 B_0\hat{x}$. So the force on a positive charge would be in $+\hat{x}$, but electrons (negative) feel a force in $-\hat{x}$, driving them toward the origin. Electrons pile up at the origin, leaving the $(L,0,0)$ end electron-deficient — meaning positively charged. Once charge separation builds up, an electrostatic field develops inside the rod to oppose further migration. At equilibrium, this induced electric field must point from the positive end toward the negative end, i.e., from $(L,0,0)$ toward the origin: the $-\hat{x}$ direction. This confirms C is correct. Choice A correctly identifies $(L,0,0)$ as positive but gets the field direction wrong — the electric field inside points from positive to negative ($-\hat{x}$), not toward the positive end. Choice B incorrectly claims the origin becomes positive by invoking force on positive ions, but free charges in a conductor are electrons, not ions. Choice D correctly identifies that electrons move toward $(L,0,0)$... except they don't — the force on electrons is in $-\hat{x}$, toward the origin, not $+\hat{x}$. A reliable strategy: always compute $\vec{v} \times \vec{B}$ first, then flip the sign for electrons. The induced electric field at equilibrium always points from positive to negative inside the rod — opposite to the charge-separation force.
2. A student argues: 'The induced current flows counterclockwise (when viewed from the $+z$ direction) because by the right-hand rule, a counterclockwise current produces a field in the $+\hat{z}$ direction inside the loop, which adds to the increasing external flux and thus satisfies Faraday's law.' Which of the following best evaluates this student's reasoning?
   Answer: The student's conclusion is incorrect: the induced current flows clockwise, producing a field in the $-\hat{z}$ direction inside the loop, which opposes the increasing $+\hat{z}$ flux as required by Lenz's law. The student correctly applied the right-hand rule but drew the wrong conclusion about what Lenz's law requires. Whenever you see a question involving Lenz's law, anchor yourself to one core principle: the induced current opposes the change in flux — not the flux itself, and certainly not in a way that adds to the change. Here, the external field $\vec{B} = B_0\hat{z}$ is suddenly switched on, meaning the flux through the loop is increasing in the $+\hat{z}$ direction. Lenz's law demands that the induced current create a magnetic field that fights this increase — so the induced field inside the loop must point in the $-\hat{z}$ direction (into the page). Using the right-hand rule: curl your fingers so your thumb points in $-\hat{z}$, and your fingers curl clockwise when viewed from $+z$. Therefore, the induced current flows clockwise, making B the correct answer. The student correctly applied the right-hand rule (a $+\hat{z}$ field does correspond to a counterclockwise current), but misidentified what Lenz's law requires, concluding the wrong direction. A is wrong because the flux is increasing, not decreasing — there's no basis for invoking "opposing decreasing flux" here, and the conclusion (counterclockwise) is also incorrect. C is wrong because it validates the student's flawed reasoning entirely. Lenz's law never says the induced field adds to the external flux — that would violate the law completely. D is wrong because it claims the student also misread the right-hand rule. In fact, the student applied the right-hand rule correctly; their only error was misunderstanding Lenz's law. Your study tip: always ask two separate questions — "Is the flux increasing or decreasing?" and "What field direction opposes that change?" Keep these steps distinct to avoid the exact trap this student fell into.

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