A metal rod PQ lies along the y-axis, with its end P at the origin and its end Q on the positive y-axis. A uniform magnetic field is directed along the +z axis. Which of the following is correct?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → If the rod is moved along the +x direction, the end P is at the higher potential.
The free charges in a moving rod move with it. Each one feels the Lorentz force \( q(\vec{v}\times\vec{B}) \). An emf appears across the rod only when this force has a part along the length of the rod. The end to which it pushes positive charge becomes the positive end. For motion along +x, the force on a positive charge acts along:
\( \begin{aligned} \vec{v}\times\vec{B} &= (v\,\hat{\imath})\times(B\,\hat{k}) \\ &= -vB\,\hat{\jmath} \end{aligned} \)
Option (1) If the rod is moved along the +y direction, an emf is induced across it and the end Q is at the higher potential. — Incorrect. Here \( (v\,\hat{\jmath})\times(B\,\hat{k}) \) points along +x, at right angles to the rod. Nothing pushes the charges along the rod, so no emf appears between P and Q.
Option (2) If the rod is moved along the +x direction, the end Q is at the higher potential. — Incorrect. The force on positive charge is along −y, from Q towards P. The free electrons are pushed the other way and collect at Q. So Q is at the lower potential.
Option (3) If the rod is moved along the +x direction, the end P is at the higher potential. — Correct. \( \vec{v}\times\vec{B} \) lies along −y, along the rod and towards P. Positive charge is driven to P, so P becomes the positive end of the rod.
Option (4) If the rod is moved along the +z direction, the end P is at the higher potential. — Incorrect. The velocity is then parallel to the field, so \( \vec{v}\times\vec{B}=0 \). No force acts on the charges and both ends stay at the same potential.
NCERT (2026-27):
Magnetic Field, Lorentz Force – “(ii) The magnetic force q [ v × B ] includes a vector product of velocity and magnetic field. The vector product makes the force due to magnetic field vanish (become zero) if velocity and magnetic field are parallel or anti-parallel. The force acts in a (sideways) direction perpendicular to both the velocity and the magnetic field. Its direction is given by the screw rule or right hand rule for vector (or cross) product ….”
Motional Electromotive Force – “… It is also possible to explain the motional emf expression … by invoking the Lorentz force acting on the free charge carriers of conductor PQ.”