Q27 of 47 Page 1

(a) Describe briefly the process of transferring the charge between the two plates of a parallel plate capacitor when connected to a battery. Derive an expression for the energy stored in a capacitor.

(b) A parallel plate capacitor is charged by a battery to a potential difference V. It is disconnected from battery and then connected to another uncharged capacitor of the same capacitance. Calculate the ratio of the energy stored in the combination to the initial energy on the single capacitor.


OR


(a) Derive an expression for the electric field at any point on the equatorial line of an electric dipole.


(b) Two identical point charges, q each, are kept 2m apart in air. A third point charge Q of unknown magnitude and sign is placed on the line joining the charges such that the system remains in equilibrium. Find the position and nature of Q.


(a)


Suppose an uncharged capacitor of capacitance is connected in series with a switch and a battery of emf . The switch is initially open. Then, the switch is closed at . Initially, the capacitor acts as a short circuit. The current flows and deposits charge in the capacitor. The plate with the positive terminal of the battery loose electron and the other plate gains electron. Due to this build-up of charges, potential difference across the capacitor build up. When this potential difference is equal to the emf of the battery, the capacitor is said to be fully charged and the capacitor acts like an open switch.


At time t, the capacitor has charge . Let the potential difference across the plates be given by:



A small charge is being transferred from one plate to another. The work in this process is given by:



Integrating, we get





Using ,



This work done is stored as energy on the capacitor.


(b)


Let the capacitance of the capacitor be .


The energy stored by the capacitor when it is charged is given by:



The charge on the capacitor is


Now, when the capacitor is connected to the other capacitor, they both get the same charge in equilibrium by symmetry. As the total charge remains the same, the charge on each capacitor is



Now, the energy stored is





The ratio is:



OR


Let charges and be separated by a distance . Let point P be any point on the equatorial line of the dipole. Let it be a distance form the line joining the two charges.



Now,



The electric field at P due to the charges is:




The vertical components will cancel out. We need to add the horizontal components. Hence,



Let be the unit vector in the direction of the dipole moment (from positive charge to negative charge).




Now, by trigonometry,





If ,



(b)


All charges must be in equilibrium. As the coulomb force depends upon distance and by symmetry, the charge Q must be in the middle of the two charges. For the forces to balance on the q’s, Q must have opposite sign to that of q.


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25

(a) In a series LCR circuit connected across an ac source of variable frequency, obtain the expression for its impedance and draw a plot showing its variation with frequency of the ac source.

(b) What is the phase difference between the voltages across inductor and the capacitor at resonance in the LCR circuit ?


(c) When an inductor is connected to a 200 V dc voltage, a current of 1A flows through it. When the same inductor is connected to a 200 V, 50 Hz ac source, only 0.5 A current flows. Explain, why ? Also, calculate the self inductance of the inductor.


OR


(a) Draw the diagram of a device which is used to decrease high ac voltage into a low ac voltage and state its working principle. Write four sources of energy loss in this device.


(b) A small town with a demand of 1200 kW of electric power at 220 V is situated 20 km away from an electric plant generating power at 440 V. The resistance of the two-wire line carrying power is 0.5 Ω per km. The town gets the power from the line through a 4000-220 V step-down transformer at a sub-station in the town. Estimate the line power loss in the form of heat.


 26

(a) Describe any two characteristic features which distinguish between interference and diffraction phenomena. Derive the expression for the intensity at a point of the interference pattern in Young’s double slit experiment.

(b) In the diffraction due to a single slit experiment, the aperture of the slit is 3 mm. If monochromatic light of wavelength 620 nm is incident normally on the slit, calculate the separation between the first order minima and the 3rd order maxima on one side of the screen. The distance between the slit and the screen is 1.5 m.


OR


(a) Under what conditions is the phenomenon of total internal reflection of light observed ? Obtain the relation between the critical angle of incidence and the refractive index of the medium.


(b) Three lenses of focal lengths +10 cm, –10 cm and +30 cm are arranged coaxially as in the figure given below. Find the position of the final image formed by the combination.



 2

When unpolarised light is incident on the interface separating the rarer medium and the denser medium, Brewster angle is found to be 60°. Determine the refractive index of the denser medium.

 4

When a potential difference is applied across the ends of a conductor, how is the drift velocity of the electrons related to the relaxation time?