A current loop $$ABCD$$ is held fixed on the plane of the paper as shown in the figure. The arcs $$BC$$ (radius $$= b$$) and $$DA$$ (radius $$=a$$) of the loop are joined by two straight wires $$AB$$ and $$CD$$. A steady current $$I$$ is flowing in the loop. Angle made by $$AB$$ and $$CD$$ at the origin $$O$$ is $${30^ \circ }.$$ Another straight thin wire steady current $${I_1}$$ flowing out of the plane of the paper is kept at the origin.

The magnitude of the magnetic field $$(B)$$ due to the loop $$ABCD$$ at the origin $$(O)$$ is :

A current loop $$ABCD$$ is held fixed on the plane of the paper as shown in the figure. The arcs $$BC$$ (radius $$= b$$) and $$DA$$ (radius $$=a$$) of the loop are joined by two straight wires $$AB$$ and $$CD$$. A steady current $$I$$ is flowing in the loop. Angle made by $$AB$$ and $$CD$$ at the origin $$O$$ is $${30^ \circ }.$$ Another straight thin wire steady current $${I_1}$$ flowing out of the plane of the paper is kept at the origin.

Due to the presence of the current $${I_1}$$ at the origin:

A mixture of light, consisting of wavelength $$590$$ $$nm$$ and an unknown wavelength, illuminates Young's double slit and gives rise to two overlapping interference patterns on the screen. The central maximum of both lights coincide. Further, it is observed that the third bright fringe of known light coincides with the $$4$$th bright fringe of the unknown light. From this data, the wavelength of the unknown light is :

In an optics experiment, with the position of the object fixed, a student varies the position of a convex lens and for each position, the screen is adjusted to get a clear image of the object. A graph between the object distance $$u$$ and the image distance $$v,$$ from the lens, is plotted using the same scale for the two axes. A straight line passing through the origin and making an angle of $${45^ \circ }$$ with the $$x$$-axis meets the experimental curve at $$P.$$ The coordinates of $$P$$ will be :