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Q.

The resistivity of copper at room temperature is ${"mathml":"<math class="image_resized" style="width:38px;height:11px"><span style="font-family:$ ${"mathml":"<math class="image_resized" style="width:76px;height:12px"><span style="font-family:$ . If the density of the mobile electrons is ${"mathml":"<math class="image_resized" style="width:98px;height:12px"><span style="font-family:$ , the relaxation time for the free electrons in copper is (mass of electron ${"mathml":"<math class="image_resized" style="width:80px;height:16px"><span style="font-family:$ and charge of electron ${"mathml":"<math class="image_resized" style="width:90px;height:12px"><pstyle="margin-top:0pt;margin-bottom:0pt;line-height:115%;font-size:12pt"><spanstyle="font-family:$

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a

b

c

d

${"mathml":"<math class="image_resized" style="width:79px;height:12px"><span style="font-family:$

answer is D.(Unlock A.I Detailed Solution for FREE)

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Detailed Solution

Concept- When we hear the term density, the word distribution ought to be the first thing that comes to mind. For instance, terms like mass density (in Science, just termed density) , population density, nuclear density, etc., all refer to the distribution.Current through a conductor flows because of the electric field

E

applied across its length. It can be calculated by,

E = V l

Where

V =

potential difference across the conductor and

l =

length of the conductor
Relaxation time

τ

is defined as the time interval between two successive collisions of electrons in a conductor when current flows through it.
Relation between drift velocity

v d

and relaxation time

τ

is given by,

v d = − e E m τ

Where,

e =

charge of electron

E =

Electric field

m =

Mass of electron
Let us assume that the length of the copper conductor through which the current is flowing is

L

, area of cross-section is

A

and its current density is

n

.

⇒ I = − n e A v d

Substituting the value of

v d

from the previous equation,

⇒ I = n e A e E m τ

⇒ I = n e 2 A E m τ

Substituting the value of

E

from fist equation,

⇒ I = n e 2 A V m l τ

⇒ V I = m L n e 2 A τ

...................(1)
Now according to Ohm’s law,

⇒ V = I R

Where,

R =

resistance of the conductor.

⇒ R = V I

...........(2)
Resistance can also be calculated by,

⇒ R = ρ l A

...............(3)
Where,

ρ =

resistivity of the conductor.
Substituting equation two in equation one,

⇒ R = m L n e 2 A τ

...........(4)
Substituting equation three in equation four,

⇒ ρ L A = m L n e 2 A τ

⇒ ρ = m n e 2 τ

⇒ τ = m n e 2 ρ

Substituting the values given in the question in the above equation,

⇒ τ = 9 × 10 − 11 8.4 × 1028 × 1.6 × 10 − 19 2 × 1.7 × 10 − 8

⇒ τ = 9 × 10 − 11 8.4 × 1028 × 2.56 × 10 − 38 × 1.7 × 10 − 8

∴ τ = 2.5 × 10 − 14 s

Hence, option 4 is the correct answer.

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