A cell of internal resistance 1 ohm, 1.5 volt e.m.f. and another cell of internal resistance 2 ohm, 2 volt e.m.f. are connected in parallel across the ends of an external resistance of 5 ohm. Find the current in each branch of the circuit.

2Medium10 marks

The raising (J+) and lowering (J−) operators are defined by J+ = Jx+iJy and J− = Jx−iJy respectively. Prove the following identities :(i) [Jz, J±] = ±ℏJ±(ii) J−J+ = J2 − Jz 2 − ℏJz

3Medium20 marks

Show that for an n-type semiconductor, the Fermi level lies midway between the donor states and the conduction band edge at low temperature (assuming Ev = 0).

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