An electron , a proton and an alpha particle having the same kinetic energy are moving in circular orbits of radii \(r_e,r_p,r_\alpha\) respectively in a uniform magnetic field \(B\). The relation between \(r_e,r_p,r_\alpha\)is:
1. \( r_e>r_p=r_\alpha \)
2. \( r_e<r_p=r_\alpha \)
3. \( r_e<r_p<r_\alpha \)
4. \( r_e<r_\alpha<r_p \)

A beam of protons with speed \(4 \times 10^5 \mathrm{~ms}^{-1}\) enters a uniform magnetic field of \(0.3\) T at an angle of \(60^\circ\) to the magnetic field. The pitch of the resulting helical path of protons is close to: (Mass of the proton = \(1.67 \times 10^{-27} \mathrm{~kg}\), charge of the proton =\(1.69\times 10^{-19}\) C )
1. \(12 ~\text{cm}\)
2. \(4 ~\text{cm}\)
3. \(5~\text{cm}\)
4. \(2 ~\text{cm}\) 

The figure shows a region of length '\(l\)' with a uniform magnetic field of \(0.3~\mathrm{T}\) in it and a proton entering the region with velocity \(4 \times 10^5 \mathrm{~ms}^{-1}\) 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 (mass of proton = \(1.67 \times 10^{-27} \mathrm{~kg}\), charge of the proton = \(1.6 \times 10^{-19} \mathrm{~C}\))

1. \(0.11~\text{m}\)
2. \(0.22~\text{m}\)
3. \(0.44~\text{m}\)
4. \(0.88~\text{m}\)

A charged particle carrying charge \(1~\mathrm{\mu C}\) is moving with velocity \((2 \hat{i}+3 \hat{j}+4 \hat{k})\) m/s. If an external magnetic field of \((5 \hat{i}+3 \hat{j}-6 \hat{k}) \times 10^{-3} ~T\), exists in the region where the particle is moving then the force on the particle is \(\vec{F} \times 10^{-9} \mathrm{~N}\). The vector \(\vec{F}\) is:
1. \( -3.0 \hat{i}+3.2 \hat{j}-0.9 \hat{k} \)
2. \( -300 \hat{i}+320 \hat{j}-90 \hat{k} \)
3. \(-30 \hat{i}+32 \hat{j}-9 \hat{k} \)
4. \( -0.30 \hat{i}+0.32 \hat{j}-0.09 \hat{k}\)

A particle of charge \(q\) and mass \(m\) is moving with  a velocity \(-v\hat{i}(v\neq0)\) towards a large screen placed in the \(\mathrm{Y\text-Z}\) plane at a distance \(d\). If there is a magnetic field \(\vec{B}=B_0\hat{k}\), the minimum value of \(v\) for which the particle will not hit the screen is: 
1. \( \frac{q d B_0}{2 m} \)
2. \( \frac{2 q d B_0}{m} \)
3. \( \frac{q d B_0}{3 m} \)
4. \( \frac{q d B_0}{m}\)

A proton, a deuteron and an \(\alpha\text-\) particle are moving with same momentum in a uniform
magnetic field. The ratio of magnetic forces acting on them is ____ and their speeds are in the ratio ____.
1. 1 : 2 : 4 and 2 : 1 : 1
2. 2 : 1 : 1 and 4 : 2 : 1
3. 4 : 2 : 1 and 2 : 1 : 1
4. 1 : 2 : 4 and 1 : 1 : 2