Question

At room temperature, copper has free electron density of $8.4 \times {10^{28}}{m^{ - 3}}$. The electron drift velocity in a copper conductor of cross-sectional area of \[{10^{ - 6}}{m^2}\] and carrying a current of 5.4 A, will be:
A) $4m{s^{ - 1}}$
B) $0.4m{s^{ - 1}}$
C) $4cm{s^{ - 1}}$
D) $0.4mm{s^{ - 1}}$

Answer 1

Hint: When some potential difference is applied across some metal, an electric field is created within the metal piece. Due to the presence of this electric field, the free electrons of the metals start moving from lower to higher potential and a current flow is created in the opposite direction. The average uniform velocity of the free electrons is known as the drift velocity of the electron for that metal.

Formula Used:
Dependence of produced current on the cross sectional area, drift velocity and number density of electron is given by:
$i = nAe{v_d}$ (1)
Where,
i is the current through the conductor,
n is the free electron density within the conductor,
e is electron charge magnitude, $e = 1.6 \times {10^{ - 19}}C$.
${v_d}$ is the drift velocity of an electron.

Complete step by step answer:
Given:
1. Free electron density of copper is $n = 8.4 \times {10^{28}}{m^{ - 3}}$.
2. Cross-sectional area of copper conductor is $A = {10^{ - 6}}{m^2}$.
3. Current through the conductor is i=5.4A.

To find: The drift velocity of the electron, ${v_d}$.

Step 1
First, rewrite the eq.(1) to find an expression of ${v_d}$in terms of other variables:
$
  i = nAe{v_d} \\
  \therefore {v_d} = \dfrac{i}{{nAe}} \\
 $ (2)

Step 2
Now, substitute the values of i, n, A, e in eq.(2) to get the value of ${v_d}$ as:
$
  {v_d} = \dfrac{{5.4A}}{{8.4 \times {{10}^{28}}{m^{ - 3}} \times {{10}^{ - 6}}{m^2} \times 1.6 \times {{10}^{ - 19}}C}} \\
   = 4 \times {10^{ - 4}}m{s^{ - 1}} = 0.4mm{s^{ - 1}} \\
 $
So, the magnitude of the drift velocity is $0.4mm{s^{ - 1}}$.

Correct answer:
The electron drift velocity in the copper conductor will be (d) $0.4mm{s^{ - 1}}$.

Note: Many students have misconceptions about the direction of drift velocity of the electrons. Many think that the current direction is the same as the direction of drift velocity. But that’s wrong. Since, electrons are negatively charged particles so they always move from lower potential to higher potential but the direction of current is assumed to be from higher to lower potential. Hence, their directions are completely opposite.