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Q.Define resistivity of a conductor. How does the resistivity of a conductor depend upon the following :

(a) Number density of free electrons in the conductor (nn)
(b) Their relaxation time (τ\tau)
CBSECBSE Class XII Board 2024Subjective· 2mImportance★★★★★
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Resistivity ρ\rho is numerically equal to the resistance of a conductor of that material having unit length and unit area of cross-section (R=ρ l/AR = \rho\,l/A) — an intrinsic material property. For a conductor, it depends inversely on both the number density of free electrons nn and their relaxation time τ\tau: ρ=mne2τ\rho = \frac{m}{n e^2 \tau}.

The Definition

For a conductor of length ll and uniform cross-sectional area AA, the resistance is R=ρ lAR = \rho\,\dfrac{l}{A}. The constant ρ\rho — the resistivity — is therefore numerically equal to the resistance of a conductor of that material of unit length and unit area of cross-section. Equivalently, in microscopic form, ρ=E/J\rho = E/J: the ratio of the applied electric field to the current density it produces. It depends only on the material and its temperature, never on the conductor's dimensions.

The Concept: Why Resistivity Exists

Resistivity isn't just a number — it's a measure of how hard it is for electrons to move through a material. Think of it like walking through a crowd. The more people there are (higher nn), the harder it is to move. But also, if people are jostling you less frequently (longer τ\tau), you can glide through more easily.

The key insight is that resistivity comes from collisions. Electrons accelerate under an electric field, but they keep bumping into atoms and impurities. Each collision resets their motion. The relaxation time τ\tau is the average time between these collisions — the "free flight" time.

The resistivity of a conductor is given by:

ρ=mne2τ\rho = \frac{m}{n e^2 \tau}

where mm is the electron mass, ee is the electron charge, nn is the number density of free electrons, and τ\tau is the relaxation time.

Step-by-Step Derivation

1. Start with current density. When an electric field EE is applied, electrons drift with an average velocity vdv_d. The current density JJ is:

J=nevdJ = n e v_d

This is intuitive: more electrons per volume (nn) and faster drift (vdv_d) means more charge crossing per second per area.

2. Relate drift velocity to the field. Between collisions, an electron accelerates under the field EE with acceleration a=eEma = \frac{eE}{m}. The average drift velocity is half the final velocity just before collision:

vd=12aτ=eE2mτv_d = \frac{1}{2} a \tau = \frac{eE}{2m} \tau

But careful — this factor of 12\frac{1}{2} comes from assuming electrons start from rest after each collision. A more rigorous statistical treatment (which you'll see in advanced courses) gives:

vd=eEmτv_d = \frac{eE}{m} \tau

The exact factor depends on the distribution of collision times, but the proportionality is what matters for our purpose.

Tip

For exam purposes, remember the standard result: vd=eEmτv_d = \frac{eE}{m} \tau. The factor of 12\frac{1}{2} is sometimes dropped because a proper average over all collision times gives exactly this form. Don't get caught in the factor — focus on the dependencies.

3. Combine to get conductivity. Substituting vdv_d into JJ:

J=ne(eEmτ)=ne2τmEJ = n e \left( \frac{eE}{m} \tau \right) = \frac{n e^2 \tau}{m} E

4. Apply Ohm's law in microscopic form. Ohm's law says J=σEJ = \sigma E, where σ\sigma is conductivity. So:

σ=ne2τm\sigma = \frac{n e^2 \tau}{m}

5. Invert to get resistivity. Resistivity ρ\rho is the reciprocal of conductivity:

ρ=1σ=mne2τ\rho = \frac{1}{\sigma} = \frac{m}{n e^2 \tau}

Dependence on nn and τ\tau

Now we can read the dependencies directly from the formula. …

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