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A charged particle is projected with a certain initial velocity into a zone of uniform electric and magnetic fields, parallel to each other, in a direction perpendicular to the common direction of the fields.What will be the nature of the path of the charged particle?

Note by Souryajit Roy
2 years, 9 months ago

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The path is helical. I can prove it mathematically as well but it is easier to deal with it analytically and more fun provided only the nature of path is required.

WLOG, take the electric and magnetic fields to be parallel the direction of positive x-axis and the particle is projected along the positive y-axis in a standard three dimensional Cartesian coordinate system. Hence the force due to the magnetic field will be in the y-z plane and moreover only due to the y and z components of the velocity. The electric field is along the x direction and hence is responsible only for the velocity in the x direction. Since the initial velocity is in the y direction and the electric field accelerates the particle in the x direction only therefore the forces due to these field affects the velocity of the particle in perpendicular direction implying that the net motion can be explained as simply the superposition of the motion only due to individual fields!

Due to electric field: Constant acceleration in the x -direction. Velocity of the particle only due to the electric field is along the x axis and linear function of time.

Due to magnetic field: Will make the particle follow a circular path in the y-z plane ( or a plane parallel to it). It is simple case of uniform circular motion.

On superposing the above motions, we get a helix. A particle performing a uniform circular motion in a plane which constantly accelerating.

Hope this helps. Sudeep Salgia · 2 years, 9 months ago

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@Sudeep Salgia thanks. Souryajit Roy · 2 years, 9 months ago

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