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When $5V$ potential difference is applied across a wire of length $0.1m$, the drift speed of electrons is $2.5 \times {10^{ - 4}}m/s$.If the electron density in the wire is $8 \times {10^{28}}{m^{ - 3}},$calculate the resistivity of the wire.

Last updated date: 16th Jun 2024
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Hint:Write the expression of drift speed in terms of potential difference, length of wire, electron density and resistivity and then put those values as given in the question into the expression of drift velocity, you will get the resistivity.

Complete step by step answer:
Before solving lets define few terms,

Drift velocity: In physics a drift velocity is the average velocity attained by charged particles, such as electrons, in a material due to an electric field. In general, an electron in a conductor will propagate randomly at the Fermi velocity, resulting in an average velocity of zero. Applying an electric field adds to this random motion a small net flow in one direction; this is the drift.

Resistivity: It is electrical resistance of a conductor of unit cross-sectional area and unit length. A characteristic property of each material, resistivity is useful in comparing various materials on the basis of their ability to conduct electric currents. High resistivity designates poor conductors.

Now we know that for a conductor $I = nAe{v_d}$ where $n$ is electron density, $A$ is area of cross-section and ${v_d}$ is drift velocity.
Using ohm's law we have
$I = \dfrac{V}{R} = nAe{v_d}$
$ \Rightarrow I = \dfrac{V}{{\dfrac{{\rho l}}{A}}}\\
\Rightarrow I = nAe{v_d}$ where $\rho $ is resistivity.
$ \Rightarrow \rho = \dfrac{V}{{ne{v_d}l}}$
Putting values we have
$\rho = \dfrac{5}{{8 \times {{10}^8} \times 1.6 \times {{10}^{ - 19}} \times 2.5 \times {{10}^{ - 4}} \times 0.1}}\\
\therefore \rho = 1.5625 \times {10^{ - 5}}\Omega m$

Hence value of resistivity is $1.5625 \times {10^{ - 5}}\Omega m$.

Note:Most of the students have confusion in between resistance and resistivity so always remember that resistivity is the characteristic property of a material and it remains same irrespective to the change in length and area of a wire while the resistance of a wire is depends on length and area of cross section of a wire.Hence it’s value get changes upon changing these parameters.