An electron, an \(\alpha \) particle, and a proton have the same kinetic energy. Which of these particles has the shortest de Broglie wave length?
Answer
For a particle, de Broglie wavelength, \(\lambda =\frac h p\) The relation between wavelength and kinetic energy of a particle is \(K=\frac{p^2}{2m}\) \(p=\frac h{\lambda }\) \(K=\frac{\left(\frac h{\lambda }\right)^2}{2m}\) \(\lambda =\frac h{\sqrt{2\mathit{mK}}}\) From the above relation it can be concluded that for the same kinetic energy, de Broglie’s wavelength of a particle is inversely proportional to its mass. Mass of a proton, \(m_p=1836m_e\) Mass of an \(\alpha \) particle, \(m_{\alpha }=4m_p\) Therefore the masses are in the order, \(m_{\alpha }>m_p>m_e\) . Hence, the \(\alpha \) particle has the shortest de Broglie’s wavelength.
Consider the differentiable function:\( f:R\xrightarrow{}\;R\) for which \( f\left(1\right)=2\;and (x+y)={2}^{x}f\left(y\right)+{4}^{y}f\left(x\right)\forall\;x,y\in\;R\) . The value of \( f\left(4\right)\) is ( )
A particle is moving three times as fast as an electron.The ratio of the de Broglie wavelength of the particle to that of theelectron is \(1.813\times 10^{-4}\) . Calculate the particle’s mass and identify the particle.
A tangent is drawn at any point on the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) . If this tangent is intersected by the tangents at the vertices at points \(P\) and \(Q\) , then which of the following is/are true.( )
A. \(S,S^\prime ,P\) and \(Q\) are concyclic.
B. \(\mathit{PQ}\) is diameter of the circle.
C. \(S,S^\prime ,P\) and \(Q\) forms rhombus.
D. \(\mathit{PQ}\) is diagonal of acute angle of the rhombus formed by \(S,S^\prime ,P\) and \(Q\) .
Let \( f(x+\frac{1}{y})+f(x-\frac{1}{y})=2f\left(x\right)f\left(\frac{1}{y}\right)\forall\;x\), \( y\in\;R\)\( y\ne\;0 and\;f\left(0\right)=0\) then the value of \( f\left(1\right)+f\left(2\right)=?\) ( )
Let \( \overrightarrow{a}\),\( \overrightarrow{b}\) and \( \overrightarrow{c}\) be three vectors having magnitudes \( 1\),\( 1\) and \( 2\) respectively. If \( \overrightarrow{a}\times \left(\overrightarrow{a}\times \overrightarrow{c}\right)+\overrightarrow{b}=0,\) then the acute angle between \( \overrightarrow{a}\) and \( \overrightarrow{c}\) is ( )
Consider the function h(x) = \( h\left(x\right)=\frac{{g}^{2}\left(x\right)}{2}+3{x}^{3}-5\), where g(x) is a continuous and differentiable function. It is given that h(x) is a monotonically increasing function and g(0) = 4. Then which of the following is not true? ( )