Two infinitely long wires each carrying the same current and pointing in $$+y$$ direction are placed in the $$x y$$-plane, at $$x=-2 \mathrm{~cm}$$ and $$x=1 \mathrm{~cm}$$. An electron is fired with speed $$u$$ from the origin making an angle of $$+45^{\circ}$$ from the $$X$$-axis. The force on the electron at the instant it is fired is

[$$B_0$$ is the magnitude of the field at origin due to the wire at $$x=1 \mathrm{~cm}$$ alone].

Two electrons, $$e_1$$ and $$e_2$$ of mass $$m$$ and charge $$q$$ are injected into the perpendicular direction of the magnetic field $$B$$ such that the kinetic energy of $$e_1$$ is double than that of $$e_2$$. The relation of their frequencies of rotation, $$f_1$$ and $$f_2$$ is

Torque required to hold a small circular coil of 10 turns, area of $$2 \times 10^{-4} \mathrm{~m}^2$$ area of carrying 0.5 A current in the middle of a long solenoid of $$10^3$$ turns per metre carrying $$3 \mathrm{~A}$$ current, with its axis perpendicular to the axis of the solenoid is

Two concentric coils each of radius equal to $$4 \pi ~\mathrm{cm}$$ are placed at right angles to each other. If $$10 \mathrm{~A}$$ and $$24 \mathrm{~A}$$ are the currents flowing through the coils respectively, then the magnetic induction at the centre of the coils will be