To solve this problem, let's start by understanding the relationship between the kinetic energy, mass, and de Broglie wavelength of the particles.
The de Broglie wavelength
λ of a particle is given by:
λ=‌where
h is Planck's constant and
p is the momentum of the particle.
The momentum
p can be expressed as:
p=√2mK where
m is the mass and
K is the kinetic energy of the particle.
Let's denote the masses and kinetic energies of particles A and B as follows:
Mass of particle
A:mAMass of particle B:
mBSince the mass of particle
A is double that of particle
B :
mA=2mB Kinetic energy of particle A:
KAKinetic energy of particle B:
KBGiven that the kinetic energy of particle
B is
‌ th that of
A :
KB=‌ Now, let's find the ratio of the de Broglie wavelengths of particles A and B. The de Broglie wavelength for particle A is:
λA=‌The de Broglie wavelength for particle B is:
λB=‌We need to find the ratio
‌ :
Substitute the given values:
Thus, the ratio of the de Broglie wavelength of
A to that of
B is:
Option C:
1:4