The magnetic field intensity at the point $O$ of a loop with current $i$,whose shape is illustrated below is

Two long current-carrying thin wires,both with current $I$,are held by insulating threads of length $L$ and are in equilibrium as shown in the figure,with threads making an angle $\theta$ with the vertical. If the wires have mass $\lambda$ per unit length,then the value of current $I$ is ($g =$ gravitational acceleration).

Match the following and find the correct pairs.

List-$I$	List-$II$
$(A)$ Fleming's left hand rule	$(i)$ Direction of induced current
$(B)$ Right hand thumb rule	(ii) Magnitude and direction of magnetic induction
$(C)$ Biot-Savart law	(iii) Direction of force due to magnetic induction
$(D)$ Fleming's right hand rule	(iv) Direction of magnetic lines due to current

The magnetic field at the centre of a circular coil of radius $r$ is $\pi$ times that due to a long straight wire at a distance $r$ from it,for equal currents. Figure shows three cases: in all cases,the circular part has radius $r$ and straight ones are infinitely long. For the same current,the $B$ field at the centre $P$ in cases $1$,$2$,and $3$ have the ratio:

When a proton is released from rest in a room,it starts with an initial acceleration $a_0$ towards west. When it is projected towards north with a speed $v_0$,it moves with an initial acceleration $3a_0$ towards west. The electric and magnetic fields in the room are

An electron (mass $9 \times 10^{-31} \ kg$ and charge $1.6 \times 10^{-19} \ C$) moving with speed $v = c/100$ $(c = 3 \times 10^8 \ ms^{-1})$ is injected into a magnetic field $\vec{B}$ of magnitude $9 \times 10^{-4} \ T$ perpendicular to its direction of motion. We wish to apply a uniform electric field $\vec{E}$ together with the magnetic field so that the electron does not deflect from its path. Then: