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GATE 2023 EE – Question 43

Electromagnetic Fields · Biot-Savart law, Ampere law, Curl, Faraday law, Lorentz force · 2 marks · Multiple select

In the figure, the electric field $\boldsymbol E$ and the magnetic field $\boldsymbol B$ point to x and z directions, respectively, and have constant magnitudes. A positive charge ‘$q$’ is released from rest at the origin. Which of the following statement(s) is/are true.

Diagram for GATE 2023 EE question 43
  1. The charge will move in the direction of z with constant velocity.
  2. The charge will always move on the y-z plane only.
  3. The trajectory of the charge will be a circle.
  4. The charge will progress in the direction of y.

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Correct answer: (B) The charge will always move on the y-z plane only.; (D) The charge will progress in the direction of y.

Explanation

As drawn, the electric field is along $\hat z$ and the magnetic field is along $\hat x$ (the labels in the statement are interchanged relative to the figure). Starting from rest, the charge is accelerated along $\boldsymbol E$ and then deflected by the $\boldsymbol v\times\boldsymbol B$ force. The motion stays in the plane perpendicular to $\boldsymbol B$ (the $y$-$z$ plane) and is a cycloid: a circular motion superposed on the $\boldsymbol E\times\boldsymbol B$ drift, which is along $\hat y$. So the charge stays in the $y$-$z$ plane (B) and progresses along $y$ (D); the path is a cycloid, not a circle, and the velocity is not constant.