A particle slides on the surface of a fixed smooth sphere starting from the  topmost point. The angle rotated by the radius through the particle, when it leaves contact with the sphere, is:
1. \(\theta=cos^{-1}\Big(\frac13\Big)\)
2. \(\theta=cos^{-1}\Big(\frac23\Big)\)
3. \(\theta=tan^{-1}\Big(\frac13\Big)\)
4. \(\theta=sin^{-1}\Big(\frac43\Big)\)

What is the radius of curvature of the parabola traced out by the projectile in the previous problem at a point where the particle velocity makes an angle \(\frac{\theta}{2}\) with the horizontal?
1. \(r=\frac{v^2\cos^2\theta}{g\cos^2\theta}\)
2. \(r=\frac{2v\sin\theta}{g\tan\theta}\)
3. \(r=\frac{v\cos\theta}{g\sin^2\frac{\theta}{2}}\)
4. \(r=\frac{3v\cos\theta}{g\cot\theta}\)

A square loop is made by a uniform conductor wire as shown in the figure,     
The net magnetic field at the centre of the loop if the side length of the square is a:
1. \(\frac{\mu_{_0}i}{2a}\)
2. zero
3. \(\frac{\mu_{_0}i^2}{a^2}\)
4. None of these

The electron of an H-atom is revolving around the nucleus in circular orbit having radius \(\frac{h^2}{4\pi me^2}\) with \(\Big(\frac{2\pi e^2}{h}\Big).\) The current produced due to the motion of the electron is:
1. \(\frac{2\pi m^2e^2}{3h^2}\)
2. zero
3. \(\frac{2\pi^2me}{h^2}\)
4. \(\frac{4\pi^2me^5}{h^3}\)

Two small balls, each carrying a charge \(q\) are suspended by equal insulator strings of length 1 m from the hook of a stand. This arrangement is carried in a satellite in space. The tension in each string will be:
1. \(\frac{1}{4\pi \varepsilon_0}\frac{q}{I^2}\)
2. \(\frac{1}{4\pi\varepsilon_0}\frac{q^2}{4I^2}\)
3. \(\frac{1}{4\pi\varepsilon_0}\frac{q^2}{I^2}\)
4. \(\frac{1}{(4\pi~\varepsilon_0)}\frac{q}{I}\)

A vessel of depth \(t\) is half filled with a liquid having a refractive index \(n_1\) and the other half is filled with water having a refractive index \(n_2.\) The apparent depth of the vessel as viewed from the top is:
1. \(\frac{2t(n_1+n_2)}{n_1n_2}\)
2. \(\frac{tn_1n_2}{(n_1+n_2)}\)
3. \(\frac{t(n_1+n_2)}{2n_1n_2}\)
4. \(\frac{n_1n_2}{(n_1+n_2)t}\)