**Free Response Question No.1**

An electron is accelerated by a positive potential *V* of the order of a few volts.

(a) Derive an expression for the de Broglie wave length *λ* of the electron in terms of the electron mass *m*, electron charge *e*, Planck’s constant *h* and the accelerating potential *V*.

(b) A photon and an electron have the same wave length *λ. *Determine their kinetic energy in term of the wave length.

(c) Explain with reason, which particle in (b) has greater kinetic energy.

(d) If you substitute the known values of the electron mass *m*, electron charge *e* and the Planck’s constant *h*, the de Broglie wave length of an electron for accelerating voltages less than 50 volts or so will work out to be very nearly equal to √(150/*V*) Ǻ where *V* is the accelerating voltage (in volt). Using this simple expression, calculate the de Broglie wave length of an electron of energy 15 electron volt.

(e) Explain why you cannot use the simple expression in (d) for calculating the de Broglie wave length if the accelerating voltage is high as for example, in an X-ray tube.

**Free Response Question No.2**

**(a)** Derive an expression for the kinetic energy of the electron in a hydrogen atom in terms of the radius ‘*r*’ of the orbit, electronic charge ‘*e*’ and the permittivity ‘*ε*_{0}’_{ }of free space. Show that the potential energy of the electron is negative and of value twice that of the kinetic energy.

**(b)** The total energy *E*_{n} of the electron in the *n*^{th} orbit in a hydrogen atom is given by *E*_{n }= – *me*^{4}/8*ε*_{0}^{2}*n*^{2}*h*^{2} where ‘*m*’ is the mass of the electron (very nearly) and ‘*h*’ is Planck’s constant. Explain how this expression will get modified in *positronium* in which a positron is to be considered in place of the proton in a hydrogen atom.

**(c)** An electron and a positron moving along a straight line in opposite directions with equal speeds undergo a head-on collision and get annihilated, producing two photons of the same energy. Explain why there must be two photons produced instead of just one.

**(d)** Explain why the two photons produced in the annihilation stated in (c) must be of the same energy.

Question No.1 carries 10 points which may be divided among (a), (b), (c), (d) and (e) as 3+2+1+2+2.

Question No.2 also carries 10 points (4+3+2+1).

Try to answer these questions. I’ll be back shortly with the answers

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