Consider the motion of a positive point charge in a region where there are simultaneous uniform electric and magnetic fields and At time , this charge has velocity in the plane, making an angle with the -axis. Which of the following option(s) is (are

Question

Consider the motion of a positive point charge in a region where there are simultaneous uniform electric and magnetic fields $\vec{E} = E_{0} \hat{j}$ and $\vec{B} = B_{0} \hat{j} .$ At time $t = 0$ , this charge has velocity $\vec{v}$ in the $x - y$ plane, making an angle $θ$ with the $x$ -axis. Which of the following option(s) is (are) correct for time $t > 0$ ? [2012]

Smart Achievers · Accepted Answer

(3, 4)

If $θ = 0 °$ , the charged particle is projected along the $x$ -axis. Due to the magnetic field $\vec{B}$ , the charged particle will tend to move in a circular path in the $y - z$ plane, but due to the force of the electric field $\vec{E}$ , the particle will move in a helical path with increasing pitch. Hence options (1) and (2) are wrong.

If $θ = 10 °$ , we can resolve velocity into two rectangular components: one along the $x$ -axis $(v \cos 10 °)$ and one along the $y$ -axis $(v \sin 10 °)$ . Due to $v \cos 10 °$ , the particle will move in a circular path and due to $v \sin 10 °$ plus the force due to the 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 the magnetic field is zero. But the force due to the electric field will accelerate the particle along the $y$ -axis.

Solution

1 If $θ = 0 °$ , the charge moves in a circular path in the $x - z$ plane.

2 If $θ = 0 °$ , the charge undergoes helical motion with constant pitch along the $y$ -axis.

3 If $θ = 10 °$ , the charge undergoes helical motion with its pitch increasing with time, along the $y$ -axis.

4 If $θ = 90 °$ , the charge undergoes linear but accelerated motion along the $y$ -axis.

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Solution

1 If θ=0°, the charge moves in a circular path in the x-z plane.

2 If θ=0°, the charge undergoes helical motion with constant pitch along the y-axis.

3 If θ=10°, the charge undergoes helical motion with its pitch increasing with time, along the y-axis.

4 If θ=90°, the charge undergoes linear but accelerated motion along the y-axis.

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1 If $θ = 0 °$ , the charge moves in a circular path in the $x - z$ plane.

2 If $θ = 0 °$ , the charge undergoes helical motion with constant pitch along the $y$ -axis.

3 If $θ = 10 °$ , the charge undergoes helical motion with its pitch increasing with time, along the $y$ -axis.

4 If $θ = 90 °$ , the charge undergoes linear but accelerated motion along the $y$ -axis.