Question

If potential difference V applied across a conductor is increased to 2V , how will the drift velocity of the electron change?

Answer 1

Short Answer

If the potential difference V applied across a conductor is increased to 2V, the drift velocity of the electrons will also double.

Detailed Explanation

Let's break down why this happens step-by-step:

1. Potential Difference and Electric Field:
   When you apply a potential difference (voltage) $V$ across a conductor of length $L$, it creates a uniform electric field $E$ inside the conductor. The relationship is:
   $E = \frac{V}{L}$
   This electric field is what "pushes" the free electrons.

2. Electric Field and Force:
   The electric field $E$ exerts a force $F$ on each free electron. Since an electron has a charge $e$, the force is:
   $F = eE$

3. Force and Acceleration:
   According to Newton's second law ($F=ma$), this force causes the electrons to accelerate. The acceleration $a$ of an electron with mass $m$ is:
   $a = \frac{F}{m} = \frac{eE}{m}$

4. Drift Velocity:
   An electron doesn't accelerate indefinitely. It constantly collides with the atoms (ions) of the conductor's lattice. These collisions randomize its direction and slow it down.
   Drift velocity ($v_d$) is the average velocity that an electron achieves due to the electric field, despite these constant collisions. It is related to the electron's acceleration ($a$) and the average time between collisions, known as the relaxation time ($\tau$).
   $v_d = a \tau$

5. Putting It All Together:
   Now, let's substitute the expressions from the previous steps into the equation for drift velocity:
   $v_d = (\frac{eE}{m}) \tau$
   And since $E = V/L$:
   $v_d = (\frac{e(V/L)}{m}) \tau$
   $v_d = (\frac{e \tau}{mL}) V$

   In this final equation, the terms in the parenthesis ($e, \tau, m, L$) are all constants for a given conductor under constant temperature.
   $e$ = charge of an electron (constant)
   $m$ = mass of an electron (constant)
   $L$ = length of the conductor (constant)
   $\tau$ = relaxation time (assumed constant if temperature doesn't change)

   Therefore, we can see a direct relationship:
   $v_d \propto V$
   The drift velocity is directly proportional to the applied potential difference.

Conclusion

• Initial State: $v_{d1} \propto V$
• New State: The potential difference is doubled to $2V$.
• Result: The new drift velocity $v{d2}$ will be:
  $v{d2} \propto 2V$
  Therefore, $v{d2} = 2 \times v{d1}$.

Doubling the potential difference doubles the electric field, which doubles the force on the electrons, which doubles their average drift velocity.