GATE 2026 PH – Question 57
Consider a particle of mass $m=9.0\times10^{-5}$ kg and charge $q=3.0\times10^{-4}$ C in a uniform electromagnetic field $\mathbf E=2\hat x\ \mathrm{V\,m^{-1}}$, $\mathbf B=3\hat z\ \mathrm{V\,m^{-2}s}$. The particle is released from the coordinates (0, 5 m, 0) at time $t=0$. Starting from initial speed zero, it comes back to the y-axis for the first time at time $t$. The value of $t$ in seconds (rounded off to two decimal places) is _____
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: 0.62 to 0.64
Explanation
In crossed electric and magnetic fields a charged particle released from rest follows a **cycloid**.
**Cyclotron frequency:**
$$\omega_c=\frac{qB}{m}=\frac{3.0\times10^{-4}\times3}{9.0\times10^{-5}}=10\ \text{rad/s}.$$
**Motion in the field plane.** The electric field is along $x$ and the magnetic field is along $z$, so the motion is in the $x$-$y$ plane. Starting from rest at $x=0$, the particle's $x$-coordinate is
$$x(t)=\frac{E}{\omega_cB}\left(1-\cos\omega_ct\right),$$
which is zero at $t=0$ and returns to zero when $\omega_ct=2\pi$. That is when the particle is back on the $y$ axis, at the cusp of the cycloid.
**First return time:**
$$t=\frac{2\pi}{\omega_c}=\frac{2\pi}{10}=\mathbf{0.63\ s}.$$