A thin rectangular magnet suspended freely has a period of oscillation equal to $$T.$$ Now it is broken into two equal halves (each having half of the original length) and one piece is made to oscillate freely in the same field. If its period of oscillation is $$T',$$ the ratio $${{T'} \over T}$$ is

A particle of mass $$M$$ and charge $$Q$$ moving with velocity $$\overrightarrow v $$ describe a circular path of radius $$R$$ when subjected to a uniform transverse magnetic field of induction $$B.$$ The network done by the field when the particle completes one full circle is

A particle of charge $$ - 16 \times {10^{ - 18}}$$ coulomb moving with velocity $$10m{s^{ - 1}}$$ along the $$x$$-axis enters a region where a magnetic field of induction $$B$$ is along the $$y$$-axis, and an electric field of magnitude $${10^4}V/m$$ is along the negative $$z$$-axis. If the charged particle continues moving along the $$x$$-axis, the magnitude of $$B$$ is

A magnetic needle lying parallel to a magnetic field requires $$W$$ units of work to turn it through $${60^ \circ }.$$ The torque needed to maintain the needle in this position will be :