A circular coil having N turns and radius r carries a current I. It is held in the XZ plane in a magnetic field B$${\mathop i\limits^ \wedge }$$ . The torque on the coil due to the magnetic field is :

A proton and an $$\alpha $$-particle (with their masses in the ratio of 1 : 4 and charges in the ratio of 1 : 2) are accelerated from rest through a potential difference V. If a uniform magnetic field (B) is set up perpendicular to their velocities, the ratio of the radii r_p : r_{$$\alpha $$} of the circular paths described by them will be ;

As shown in the figure, two infinitely long, identical wires are bent by 90^o and placed in such a way that the segments LP and QM are along the x-axis, while segments PS and QN are parallel to the y-axis. If OP = OQ = 4cm, and the magnitude of the magnetic field at O is 10^–4 T, and the two wires carry equal currents (see figure), the magnitude of the current in each wire and the direction of the magnetic field at O will be ($$\mu $$₀ = 4$$\pi $$ $$ \times $$ 10^–7 NA^–2) :

The region between y = 0 and y = d contains a magnetic field $$\overrightarrow B = B\widehat z$$. A particle of mass m and charge q enters the region with a velocity $$\overrightarrow v = v\widehat i.$$ If d $$=$$ $${{mv} \over {2qB}},$$ the acceleration of the charged particle at the point of its emergence at the other side is :