De-Broglie Wavelength and its Formula

De-Broglie wavelength is the wavelength associated with a moving particle. It is given by the formula:

λ = h/p

Where λ is the De-Broglie wavelength, h is Planck's constant, and p is the momentum of the particle.

Ratio of De-Broglie Wavelengths

The ratio of De-Broglie wavelengths of alpha particles accelerated through potentials of 100 V and 8100V can be found using the formula:

λ1/λ2 = √(V2/V1)

Where λ1 and λ2 are the De-Broglie wavelengths of the alpha particle accelerated through potentials of 100 V and 8100 V respectively, and V1 and V2 are the potentials applied.

Calculations
Given, V1 = 100 V and V2 = 8100 V

The momentum of the alpha particle can be calculated using the formula:

p = √(2mK)

Where m is the mass of the alpha particle and K is the kinetic energy.

For V1 = 100 V, K = eV1 = 1.6 x 10^-19 x 100 J = 1.6 x 10^-17 J
p1 = √(2 x 6.64 x 10^-27 x 1.6 x 10^-17) kg m/s = 3.52 x 10^-22 kg m/s
λ1 = h/p1 = 6.626 x 10^-34 J s / 3.52 x 10^-22 kg m/s = 1.88 x 10^-12 m

For V2 = 8100 V, K = eV2 = 1.6 x 10^-19 x 8100 J = 1.296 x 10^-15 J
p2 = √(2 x 6.64 x 10^-27 x 1.296 x 10^-15) kg m/s = 1.05 x 10^-20 kg m/s
λ2 = h/p2 = 6.626 x 10^-34 J s / 1.05 x 10^-20 kg m/s = 6.30 x 10^-15 m

Therefore, the ratio of De-Broglie wavelengths is:

λ1/λ2 = √(V2/V1) = √(8100/100) = 9

Hence, the ratio of De-Broglie wavelengths of alpha particles accelerated through potentials of 100 V and 8100 V is 9.