Dual Nature Of Matter

de-Broglie wavelength: Kinetic energy: Substitute values:

Let be the common wavelength. For the photon:
For the particle: de Broglie wavelength
Kinetic energy of the particle:
So, ratio:
Now substitute , so
Put back in expression:
Simplify and substitute to get: ⇒ Ratio = 3 : 8

Step 1: Use relation for momentum from motion in magnetic field.
Step 2: Use de Broglie wavelength formula.
Step 3: Insert known values.
Step 4: Select the correct option.
The de Broglie wavelength is approximately 2.1 \AA, which matches option (4).

Step 1: Identify the Formula for de Broglie Wavelength
The de Broglie wavelength ( ) of a particle is given by the formula: Where is Planck's constant and is the momentum of the particle. The momentum ( ) of a particle with mass ( ) and speed ( ) is given by . Substituting this into the de Broglie wavelength formula: Step 2: Extract Given Values and Ensure SI Units
Given: \begin{itemize} \item Mass of the particle, \item Speed of the particle, \item Planck's constant, \end{itemize} All given values are already in their standard SI units (kg, m/s, J s), so no unit conversion is required at this stage. Step 3: Compute the de Broglie Wavelength ( )
Substitute the given values into the de Broglie wavelength formula: First, calculate the denominator (momentum): Now, calculate the wavelength: To express this in angstroms ( ), recall that . Step 4: Analyze Options
\begin{itemize} \item Option (1): 200 . Incorrect. \item Option (2): 0.2 . Incorrect. \item Option (3): 2 . Correct, as it matches our calculated wavelength. \item Option (4): 20 . Incorrect. \end{itemize}