State the conclusions of Rutherford's \( \alpha \)-particle scattering experiment.
Rutherford's model of the atom replaced Thomson's "plum pudding" model by introducing the nucleus.
Rutherford's experiment concluded the existence of a small, dense, and positively charged nucleus at the center of the atom.
Key conclusions from the \( \alpha \)-particle scattering experiment:
1. Most \( \alpha \)-particles passed through the gold foil, indicating atoms are mostly empty space.
2. A few \( \alpha \)-particles were deflected at small angles, suggesting a positively charged region in the atom.
3. Very few were deflected back, confirming that the nucleus is small, massive, and positively charged.
Two parallel plate capacitors of capacitances \( C \) and \( 2C \) are joined with a battery of voltage difference \( V \) as shown in the figure. If the battery is removed and the space between the plates of the capacitor of capacitance \( C \) is completely filled with a material of dielectric constant \( K \), then find out:
Define the coefficient of mutual inductance.
The coefficient of mutual inductance (\(M\)) is a measure of the ability of one coil to induce an electromotive force (EMF) in a nearby coil due to a changing current in the first coil. It is given by: \[ e = -M \frac{di}{dt} \]
If a current \( i = 10 \sin(100 \pi t) \, \mathrm{A} \) is flowing in a primary coil, and the maximum induced electromotive force in the secondary coil placed near it is \( 5 \pi \, \mathrm{volt} \), then the coefficient of mutual induction between these coils needs to be determined.
Differentiate between interference and diffraction of light. Explain qualitatively the diffraction phenomenon of light by a single slit. Light of 6000 Ã… wavelength is incident normally on a single slit of width \( 3 \times 10^{-4} \, \text{cm} \). Find out the angular width of the central maxima.
Show that the circumference of the orbit of an electron revolving in the \( n \)-th orbit is equal to \( n\lambda \) with the help of Bohr's quantum theory. Also, show the emission and absorption spectral lines between energy levels \( n = 1 \) and \( n = 3 \) of hydrogen atom.
What do you mean by the current sensitivity of a moving coil galvanometer? Resistance of a galvanometer is \( 50 \, \Omega \) and for full-scale deflection, the current is \( 0.05 \, \mathrm{A} \). What would be the required length of a wire to convert it into an ammeter of 5 A range? (Area of cross-section of wire = \( 2.7 \times 10^{-6} \, \mathrm{m^2} \), specific resistance of the wire material = \( 5.0 \times 10^{-7} \, \Omega \cdot \mathrm{m} \))
Obtain the formula for the internal resistance of a cell in terms of \( E \), \( V \), and \( R \), where \( E \), \( V \), and \( R \) are the electromotive force of the cell, potential difference across the external resistance, and external resistance, respectively.