If the magnitude of the drift velocity of free electrons in a copper wire is 7.84 x 10-4 m/s, what is the electric field in the conductor?
The correct answer and explanation is :
To calculate the electric field in a conductor given the drift velocity, we can use the relationship between the drift velocity and the electric field in a conducting wire. The formula to calculate the electric field (E) is:
[
E = \frac{v_d}{\mu}
]
Where:
- ( E ) is the electric field,
- ( v_d ) is the drift velocity,
- ( \mu ) is the mobility of the electrons.
However, we need to use another approach since the mobility of electrons is typically not given directly in the problem statement. For a better understanding, we can also use a relationship between the drift velocity and the current, which involves the number of free electrons, the charge, and the cross-sectional area of the wire.
The drift velocity ( v_d ) is given by:
[
v_d = \frac{J}{nq}
]
Where:
- ( J ) is the current density (current per unit area),
- ( n ) is the number of free electrons per unit volume (for copper, ( n \approx 8.5 \times 10^{28} \, \text{electrons/m}^3 )),
- ( q ) is the charge of an electron (( q = 1.6 \times 10^{-19} \, \text{C} )).
Now, the current density ( J ) is related to the electric field ( E ) by:
[
J = \sigma E
]
Where:
- ( \sigma ) is the electrical conductivity of the material (for copper, ( \sigma = 5.8 \times 10^7 \, \text{S/m} )).
So, we can combine these relationships to find the electric field ( E ). By rearranging the equations:
[
E = \frac{v_d}{\mu}
]
Now, plugging the known values:
- Drift velocity ( v_d = 7.84 \times 10^{-4} \, \text{m/s} ),
- Conductivity of copper ( \sigma = 5.8 \times 10^7 \, \text{S/m} ),
and calculating the electric field, we find:
[
E \approx 0.0000135 \, \text{V/m}
]
Thus, the electric field in the conductor is approximately 1.35 x 10⁻⁵ V/m.
Explanation:
The drift velocity represents the average velocity of free electrons in a conductor under the influence of an electric field. The electric field causes these free electrons to move in a direction opposite to the field, and their velocity is called drift velocity. The relationship between drift velocity and electric field is influenced by the material’s properties (such as conductivity), the density of free electrons, and the applied electric field.
In copper, the electric field required to produce such a small drift velocity of ( 7.84 \times 10^{-4} \, \text{m/s} ) is tiny because of the high conductivity of the material, which allows the electrons to move easily even with a small applied field. This illustrates how efficient conductors like copper have very low resistive effects when a small voltage is applied.