An electron gun is placed inside a long solenoid of radius R on its axis. The solenoid has n turns/length and carries a current I. The electron gun shoots an electron along the radius of the solenoid with speed v. If the electron does not hit the surface of the solenoid, maximum possible value of v is (all symbols have their standard meaning) :

A long, straight wire of radius a carries a current distributed uniformly over its cross-section. The ratio of the magnetic fields due to the wire at distance $${a \over 3}$$ and 2$$a$$, respectively from the axis of the wire is :

A charged particle of mass 'm' and charge 'q' moving under the influence of uniform electric field $$E\overrightarrow i $$ and a uniform magnetic field $$B\overrightarrow k $$ follows a trajectory from point P to Q as shown in figure. The velocities at P and Q are respectively, $$v\overrightarrow i $$ and $$ - 2v\overrightarrow j $$ . Then which of the following statements (A, B, C, D) are the correct ?
(Trajectory shown is schematic and not to scale) :
(A) E = $${3 \over 4}\left( {{{m{v^2}} \over {qa}}} \right)$$

(B) Rate of work done by the electric field at P is $${3 \over 4}\left( {{{m{v^3}} \over a}} \right)$$

(C) Rate of work done by both the fields at Q is zero

(D) The difference between the magnitude of angular momentum of the particle at P and Q is 2mav.

A very long wire ABDMNDC is shown in figure carrying current I. AB and BC parts are straight, long and at right angle. At D wire forms a circular turn DMND of radius R. AB, BC parts are tangential to circular turn at N and D. Magnetic field at the centre of circle is :