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 wires have mass $$\lambda $$ per unit-length then the value of $$I$$ is :
($$g=$$ $$gravitational$$ $$acceleration$$ )

A rectangular loop of sides $$10$$ $$cm$$ and $$5$$ $$cm$$ carrying a current $$1$$ of $$12A$$ is placed in different orientations as shown in the figures below :

If there is a uniform magnetic field of $$0.3$$ $$T$$ in the positive $$z$$ direction, in which orientations the loop would be in $$(i)$$ stable equilibrium and $$(ii)$$ unstable equilibrium ?

A conductor lies along the $$z$$-axis at $$ - 1.5 \le z < 1.5\,m$$ and carries a fixed current of $$10.0$$ $$A$$ in $$ - {\widehat a_z}$$ direction (see figure). For a field $$\overrightarrow B = 3.0 \times {10^{ - 4}}\,{e^{ - 0.2x}}\,\,{\widehat a_y}\,\,T,$$ find the power required to move the conductor at constant speed to $$x=2.0$$ $$m$$, $$y=0$$ $$m$$ in $$5 \times {10^{ - 3}}s.$$ Assume parallel motion along the $$x$$-axis.

A charge $$Q$$ is uniformly distributed over the surface of non-conducting disc of radius $$R.$$ The disc rotates about an axis perpendicular to its plane and passing through its center with an angular velocity $$\omega .$$ As a result of this rotation a magnetic field of induction $$B$$ is obtained at the center of the disc. If we keep both the amount of charge placed on the disc and its angular velocity to be constant and very the radius of the disc then the variation of the magnetic induction at the center of the disc will be represented by the figure :