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Q.(a) Describe the microscopic model of current and obtain microscopic form of Ohm's Law. OR

(b) Derive an expression for Radius and Velocity of an electron in the nth^{th} orbit using Bohr atom model.
Tamil Nadu DgeTamil Nadu HSC (DGE) Board 2023Subjective· 5mImportance★★★★★
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(a) The microscopic (free-electron) model of conduction, using drift velocity and relaxation time, derives Ohm's law in the form J=σEJ=\sigma E; (b) Bohr's postulates give the radius and velocity of the electron in the nthn^{th} orbit of a hydrogen-like atom. Both alternatives answered below.

(a) Microscopic model of current and Ohm's Law

1. Free-electron picture. In a metallic conductor, a large number of free (conduction) electrons move randomly at high thermal speeds, colliding frequently with the fixed positive ions, with zero net velocity in the absence of an applied field (random directions cancel on average).

2. Effect of an applied field. When an electric field EE is applied, each free electron (charge −e-e, mass mm) experiences a force −eE-eE, producing an acceleration a=eEma = \dfrac{eE}{m} (magnitude) between collisions. Because collisions randomise the velocity gained, the electrons acquire a small net "drift" superimposed on their random thermal motion. If τ\tau is the average time between successive collisions (relaxation time), the average drift velocity gained is

vd=aτ=eEτmv_d = a\tau = \dfrac{eE\tau}{m}

3. Relating drift velocity to current. Consider a conductor of cross-sectional area AA with free-electron number density nn (electrons per unit volume). In time dtdt, electrons drift a distance vd dtv_d\,dt, so the charge crossing area AA is dq=n e A vd dtdq = n\,e\,A\,v_d\,dt, giving current

I=dqdt=neAvdI = \dfrac{dq}{dt} = neAv_d

4. Substituting the drift velocity.

I=neA(eEτm)=ne2AτmEI = neA\left(\dfrac{eE\tau}{m}\right) = \dfrac{ne^2A\tau}{m}E

5. Microscopic (local) form of Ohm's Law. Dividing by area to get current density J=I/AJ = I/A,

J=ne2τmE=σEJ = \dfrac{ne^2\tau}{m}E = \sigma E

where σ=ne2τm\sigma = \dfrac{ne^2\tau}{m} is defined as the electrical conductivity of the material (and its reciprocal, ρ=1/σ=mne2τ\rho=1/\sigma=\dfrac{m}{ne^2\tau}, the resistivity). This relation J=σEJ=\sigma E is the microscopic statement of Ohm's Law: current density is directly proportional to the applied electric field, with the conductivity determined entirely by the material's free-electron density and relaxation time.

(b) Radius and velocity of the electron in the nthn^{th} Bohr orbit

1. Bohr's postulates used.

  • The electron revolves in a circular orbit around the nucleus (charge +Ze+Ze), the Coulomb attraction supplying the necessary centripetal force.
  • The angular momentum of the electron is quantised: mvr=nh2πmvr = \dfrac{nh}{2\pi}, n=1,2,3,…n=1,2,3,\dots

2. Force equation. Equating Coulomb force to centripetal force for an electron of mass mm, charge −e-e, speed vv, in orbit of radius rr around a nucleus of charge +Ze+Ze: …

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