Consider the motion of a positive point charge in a region where there are simultaneous uniform electric and magnetic fields $\vec{E} = E_{0} \hat{j} a n d \vec{B} = B_{0} \hat{j} .$ At time t =0, this charge has velocity $\vec{υ}$ in the in the x-y plane, making an angle θ with the x-axis. Which of the following option(s) is (are) correct for time t > 0 ?

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If $θ = 90^{o}$ the charge undergoes linear but accelerated motion along the y-axis

If $θ = 0^{o}$ , the charge undergoes helical motion with constant pitch along the y-axis.

If $θ = 0^{o}$ , the charge moves in a circular path in the x-z plane.

If $θ = 10^{o}$ , the charge undergoes helical motion with its pitch increasing with time, along the y-axis.

answer is C, D.

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Detailed Solution

If $θ = 0^{o}$ , the charged particle is projected along x- axis, due to magnetic field, $\vec{B}$ the charged particle will tend to move in a circular path in y-z plane but due to force of electric field $\vec{E}$ , the particle will move in a helical path with increasing pitch. Hence options (A) and (B) are wrong. If $θ = 10^{o}$ , we can resolve velocity into two rectangular components. One along x-axis

(v cos 10°) and one along y- axis (v sin 10°). Due to v cos 10°, the particle will move in circular path and due to v sin 10° plus the force due to electric field, the particle will undergo helical motion with its pitch increasing. If θ = 90°, the charge is moving along the magnetic field. Therefore the force due to magnetic field is zero. But the force due to electric field will accelerate the particle along y-axis.

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