The drift velocity of electrons in a wire is directly proportional to the current flowing through it. When the length of the wire conductor is doubled while keeping the potential difference constant, the resistance of the wire changes.
According to Ohm's law, the resistance (R) of a conductor is given by:
R = (ρ * L) / A
where ρ is the resistivity of the material, L is the length of the conductor, and A is the cross-sectional area of the conductor.
Since the potential difference (V) across the conductor is kept constant, the current (I) flowing through the conductor is inversely proportional to its resistance:
I = V / R
If the length of the wire is doubled, the resistance becomes:
R' = (ρ * 2L) / A = 2 * (ρ * L) / A = 2R
Therefore, the resistance doubles when the length of the wire is doubled while keeping the potential difference constant.
The drift velocity of electrons (v_d) is given by:
v_d = I / (n * A * e)
where n is the number density of electrons, A is the cross-sectional area of the wire, and e is the charge of an electron.
Since the current (I) is inversely proportional to the resistance, and the resistance is doubled, the current will be halved:
I' = I / 2
Substituting this into the equation for drift velocity:
v_d' = (I / 2) / (n * A * e) = I / (2 * n * A * e) = v_d / 2
Therefore, when the length of the wire conductor is doubled while keeping the potential difference constant, the drift velocity of electrons is halved.