The Question is from NEET guide , and it was created by the author SUBHANI

Question

An electron with initial de-Broglie wavelength $\lambda_0$ is moving with velocity $V_0 \hat{i}$ and enters a region with electric field $-E_0 \hat{i}$. How much time will it take for its de-Broglie wavelength to become $2\lambda_0$?

SUBHANI · Accepted Answer

## CORE INFORMATION
✓ **Answer**: Option C
## CONCEPT FRAMEWORK
 **Primary Concept**: de Broglie wavelength and motion of charged particles in an electric field.
 **Related Topics**: Classical mechanics, electromagnetism, quantum mechanics.
 **Prerequisites**: Understanding of de Broglie wavelength ($\lambda = \frac{h}{p}$), kinetic energy ($KE = \frac{1}{2}mv^2$), and the work-energy theorem.
## SOLUTION PATHWAY
**Given Information**:
  Initial de Broglie wavelength: $\lambda_0 = \frac{h}{mv_0}$ 
  Initial velocity: $v_0$ 
  Electric field: $-E_0\hat{i}$
  Final de Broglie wavelength: $2\lambda_0 = \frac{h}{mv}$ (where $v$ is the final velocity)
  Charge of electron: $-e$
**Method & Steps**:
 1. Relate initial and final de Broglie wavelengths:
  $\frac{h}{mv} = 2\frac{h}{mv_0} \implies v = \frac{v_0}{2}$ 
 2. Apply the work-energy theorem:
  The work done by the electric field is equal to the change in kinetic energy of the electron.
  $W = \Delta KE =  \frac{1}{2}m(\frac{v_0}{2})^2 - \frac{1}{2}mv_0^2 = -\frac{3}{8}mv_0^2$
  The work done by the electric field is also given by $W = qEd = -eE_0d$, where $d$ is the distance traveled by the electron.
  Therefore, $-eE_0d = -\frac{3}{8}mv_0^2 \implies d = \frac{3mv_0^2}{8eE_0}$
 3. Find the time taken:
  The time taken is given by $t = \frac{d}{v_{avg}}$, where $v_{avg}$ is the average velocity.
  Since the acceleration is constant, $v_{avg} = \frac{v_0 + v}{2} = \frac{v_0 + \frac{v_0}{2}}{2} = \frac{3v_0}{4}$
  $t = \frac{\frac{3mv_0^2}{8eE_0}}{\frac{3v_0}{4}} = \frac{mv_0}{2eE_0}$
  Result: $t = \frac{mv_0}{2eE_0}$
  Verification: The units of the expression are consistent with time (kg m/s)/(C N/C) = s.
 ## OPTION ANALYSIS
A :x: Incorrect calculation of the average velocity and distance traveled.
B :x: Incorrect application of the work-energy theorem and the relationship between de Broglie wavelength and velocity.
C ✓: Correct application of the work-energy theorem, average velocity calculation, and the relation between de Broglie wavelength and velocity.
D :x: Incorrectly calculates the final velocity and doesn't consider the average velocity.
 ## LEARNING AIDS
 **Common Mistakes**:
  1.Incorrectly using the average velocity: Using initial or final velocity instead of the average velocity.
  2.Incorrectly applying the work-energy theorem: Forgetting the negative sign for the work done by the electric field or incorrectly calculating the change in kinetic energy.
 **Quick Tips**:
  1.Remember that the average velocity for constant acceleration is the average of the initial and final velocities.
  2.Always check the units of your final answer to ensure consistency.

SUBHANI · Answer

$\frac{2mv_0}{eE_0}$

SUBHANI · Answer

$\frac{2h}{\lambda_0 eE_0}$

SUBHANI · Answer

The correct Option is "$\frac{mv_0}{2eE_0}$"

SUBHANI · Answer

$\frac{mv_0}{eE_0}$

NEET Questions

Select Subject

Question Tags

Question Tags

Question Tags

Question Tags

Question Tags

Question Tags

Question Tags

Question Tags

Question Tags

Question Tags

NEET Subjects

Our Bots

Company