Question

If the kinetic energy of a particle is increased by

2 %

find the percentage change in its de Broglie wavelength

(λ)

.

Accepted Answer

de Broglie wavelength of a matter wave is inversely proportional to the momentum of the particle.

Mathematically, the de Broglie wavelength $(λ)$ of a wave is related to the momentum $(p)$ of the corresponding particle is given by

$λ = \frac{h}{p} . . . . . . . . . . . . . . . . . . . . . . . . (1)$

The initial kinetic energy $(K_{1})$ of the particle can be written as

$K_{1} = \frac{{p_{1}}^{2}}{2 m} . . . . . . . . . . . . . . . . . . . . . (2)$

where, $p_{1}$ is the initial momentum and $m$ is the mass of the particle.

The final kinetic energy $(K_{2})$ is given by

$K_{2} = \frac{{p_{2}}^{2}}{2 m} . . . . . . . . . . . . . . . . . . . . . . . (3)$

where, $p_{2}$ is the final momentum.

According to the problem,

$\begin{array}{rcl} \frac{K_{2} - K_{1}}{K_{1}} & = & 0.02 \\ K_{2} & = & 1.02 K_{1} . . . . . . . . . . . . . . . . . . . . . . . (4) \end{array}$

Substitute the expressions for the kinetic energies from equations (2) and (3) into equation (3) and simplify to obtain the relation between the final and initial momentum of the particle.

$\begin{array}{rcl} \frac{{p_{2}}^{2}}{2 m} & = & 1.02 \frac{{p_{1}}^{2}}{2 m} \\ p_{2} & = & \sqrt{1.02} p_{1} . . . . . . . . . . . . . . . . . . . . . . . (5) \end{array}$

From equations (1) and (5) obtain the expressions for the initial and final de Broglie wavelengths $λ_{1}$ and $λ_{2}$ .

$λ_{1} = \frac{h}{p_{1}} . . . . . . . . . . . . . . . . . . . . . . . . . . (6)$

$λ_{2} = \frac{h}{p_{2}} = \frac{h}{\sqrt{1.02} p_{1}} . . . . . . . . . . . . . . . . . . . . . (7)$

The formula to calculate the percentage error $(P)$ in the wavelengths is given by

$P = (\frac{λ_{1} - λ_{2}}{λ_{1}} \times 100) % . . . . . . . . . . . . . . . . . . . . . . . (8)$

Substitute the expressions from equations (6) and (7) into equation (8) and solve to calculate the required percentage error.

$\begin{array}{rcl} P & = & (\frac{\frac{h}{p_{1}} - \frac{h}{\sqrt{1.02} p_{1}}}{\frac{h}{p_{1}}} \times 100) % \\ \approx & 1 % \end{array}$

	List-I		List-II
A	Franck-Hertz Experiment	(i)	Particle nature of light
B	Photo-electric experiment	(ii)	Discrete energy levels of the atom
C	Davison-Germer Experiment	(iii)	Wave nature of electron
		(iv)	Structure of atom

If the kinetic energy of a particle is increased by $2 %$ find the percentage change in its de Broglie wavelength $(λ)$ .

Important Questions on Dual Nature of Radiation and Matter

Learn and Prepare for any exam you want !

If the kinetic energy of a particle is increased by 2% find the percentage change in its de Broglie wavelength λ.

Important Questions on Dual Nature of Radiation and Matter

Learn and Prepare for any exam you want !

If the kinetic energy of a particle is increased by $2 %$ find the percentage change in its de Broglie wavelength $(λ)$ .