The figure shows a region of length $'l'$ with a uniform magnetic field of $0.3\, T$ in it and a proton entering the region with velocity $4 \times 10^{5}\, ms ^{-1}$ making an angle $60^{\circ}$ with the field. If the proton completes $10$ revolution by the time it cross the region shown, $l$ is close to....... $m$

(mass of proton $=1.67 \times 10^{-27} \,kg ,$ charge of the proton $\left.=1.6 \times 10^{-19}\, C \right)$

An electron with energy $0.1\,ke\,V$ moves at right angle to the earth's magnetic field of $1 \times 10^{-4}\,Wbm ^{-2}$. The frequency of revolution of the electron will be. (Take mass of electron $=9.0 \times 10^{-31}\,kg$ )

A car of mass $1000\,kg$ negotiates a banked curve of radius $90\,m$ on a fictionless road. If the banking angle is $45^o$, the speed of the car is ......... $ms^{-1}$

In a certain region static electric and magnetic fields exist. The magnetic field is given by $\vec B = {B_0}\left( {\hat i + 2\hat j - 4\hat k} \right)$. If a test charge moving with a velocity $\vec v = {v_0}\left( {3\hat i - \hat j + 2\hat k} \right)$ experiences no force in that region, then the electric field in the region, in $SI\, units$, is

If the magnetic field is parallel to the positive $y-$axis and the charged particle is moving along the positive $x-$axis (Figure), which way would the Lorentz force be for

$(a)$ an electron (negative charge),

$(b)$ a proton (positive charge).

An electron and a proton enter region of uniform magnetic field in a direction at right angles to the field with the same kinetic energy. They describe circular paths of radius ${r_e}$ and ${r_p}$ respectively. Then