A charged particle enters a uniform magnetic field with velocity vector making an angle of $30^o$ with the magnetic field. The particle describes a helical trajectory of pitch $x$ . The radius of the helix is
$\frac{x}{{2\pi }}$
$\frac{x}{{2\sqrt 2 \pi }}$
$\frac{x}{{2\sqrt 3 \pi }}$
$\frac{{\sqrt 3 x}}{{2\pi }}$
A rectangular region $A B C D$ contains a uniform magnetic field $B_0$ directed perpendicular to the plane of the rectangle. A narrow stream of charged particles moving perpendicularly to the side $AB$ enters this region and is ejected through the adjacent side $B C$ suffering a deflection through $30^{\circ}$. In order to increase this deflection to $60^{\circ}$, the magnetic field has to be
If an electron and a proton having same momenta enter perpendicular to a magnetic field, then
A proton (mass $m$ and charge $+e$) and an $\alpha -$ particle (mass $4m$ and charge $+2e$) are projected with the same kinetic energy at right angles to the uniform magnetic field. Which one of the following statements will be true
A proton and an $\alpha -$ particle (with their masses in the ratio of $1 : 4$ and charges in the ratio of $1:2$ are accelerated from rest through a potential difference $V$. If a uniform magnetic field $(B)$ is set up perpendicular to their velocities, the ratio of the radii $r_p : r_{\alpha }$ of the circular paths described by them will be
A positively charged particle moving due east enters a region of uniform magnetic field directed vertically upwards. The particle will