Question

How is \(v_d\) affected when the length of the conductor is doubled, keeping the voltage applied across the conductor constant?

Hint:

Be careful! Look at what is constant. - If Current (\(I\)) is constant, doubling length does *not* change \(v_d\) (since \(I = neAv_d\)). - If Voltage (\(V\)) is constant, doubling length halves the electric field (\(E = V/L\)), which explicitly halves the drift velocity (\(v_d \propto E\)). Always identify the constant parameter first!

Answer 1

Concept: The drift velocity (\(v_d\)) of free electrons inside a conducting material can be expressed microscopically in terms of the applied internal electric field strength (\(E\)) using the expression: \[ v_d = \frac{e \cdot E}{m} \cdot \tau \] Where:
• \(e\) is the fundamental charge of an electron.
• \(m\) is the rest mass of an electron.
• \(\tau\) is the average relaxation time between successive collisions of the electron with the lattice ions.
• \(E\) is the uniform electric field strength established within the conductor. For a uniform conductor of length \(L\) subjected to a constant potential difference (voltage) \(V\) across its ends, the uniform electric field strength is given by the potential gradient relation: \[ E = \frac{V}{L} \]

Step 1: Expressing drift velocity in terms of voltage and length
Substitute the potential gradient formula \(E = \frac{V}{L}\) directly into the microscopic drift velocity equation: \[ v_d = \frac{e \cdot \left(\frac{V}{L}\right)}{m} \cdot \tau = \frac{e \cdot V \cdot \tau}{m \cdot L} \] Let us isolate the constants from the variables in this problem. The problem explicitly states that the applied voltage \(V\) is kept strictly constant. Furthermore, for a given conductor at a stable constant temperature, the parameters \(e\), \(m\), and \(\tau\) are invariant physical properties. Therefore, the entire term \(\left(\frac{e V \tau}{m}\right)\) acts as a constant scaling factor. This allows us to write a direct proportionality relation: \[ v_d \propto \frac{1}{L} \] This mathematically indicates that under constant voltage conditions, the drift velocity of free electrons is inversely proportional to the total length of the conductor.

Step 2: Evaluating the effect of doubling the length
Let the initial state of the conductor be characterized by length \(L_1 = L\) and initial drift velocity \(v_{d1} = v_d\). According to our formula: \[ v_{d1} = \frac{e V \tau}{m L} \] Now, let the length of the conductor be modified to a new value \(L_2\) such that it is exactly doubled: \[ L_2 = 2L \] Let the new resulting drift velocity be denoted as \(v_{d2}\). Writing the expression for this new state: \[ v_{d2} = \frac{e V \tau}{m L_2} \] Substitute \(L_2 = 2L\) into this new equation: \[ v_{d2} = \frac{e V \tau}{m \cdot (2L)} = \frac{1}{2} \cdot \left( \frac{e V \tau}{m L} \right) \] Notice that the term enclosed in the parentheses is exactly equal to our initial drift velocity \(v_{d1}\). Substituting \(v_{d1}\) back into the expression yields: \[ v_{d2} = \frac{1}{2} \cdot v_{d1} = \frac{v_d}{2} \] Therefore, when the total length of the conductor is exactly doubled while maintaining a constant voltage across its terminals, the average drift velocity of the free electrons is reduced to exactly half of its initial value. This matches option (B).