The cut-off wavelength of the characteristic x - rays is
\({ \lambda }_{ ° }\frac { 12400 }{ V } \mathring { A } =\frac { 12400 }{ 20000 } \mathring { A }\)
= 0.62 \(\mathring { A } \)
The corresponding frequency is
\({ \upsilon }_{ o }=\frac { c }{ { \lambda }_{ o } } =\frac { 3\times 10^{ 8 } }{ 0.62\times 10^{ -10 } }\) = 4.84 x 10¹⁸ Hz

\(P=4 \ \mathrm{cal} \mathrm{} \mathrm{cm}^{-2} \mathrm{~min}^{-1}=4 \times 4.2=16.8 \mathrm{~J} \mathrm{~cm}^{-2} \mathrm{~min}^{-1} \)
\(E=\frac{h c}{\lambda}=\frac{6.626 \times 10^{-34} \times 3 \times 10^{8}}{5500 \times 10^{-10}}=3.6 \times 10^{-19} \mathrm{~J} \)
\(n=\frac{E}{hv}=\frac{16.8}{3.6 \times 10^{-19}}=4.67 \times 10^{19} \)
\(n=4.67 \times 10^{19} \)  per cm² per minute.

An alpha particle contains 2 protons and 2 neutrons. Therefore, the mass M of the alpha particle is 4 times that of a proton (m_p) (or a neutron) and its charge q is twice that of a proton (+e).
The de Broglie wavelength associated with it is
\(\lambda=\frac { h }{ \sqrt { 2MqV } } =\frac { h }{ \sqrt { 2\times (4m_{ p })\times (2e)\times V } } \)
\(=\frac { 6.626\times { 10 }^{ -34 } }{ \sqrt { 2\times 4\times 1.67\times 10^{ -27 }\times 2\times 1.6\times { 10 }^{ -19 }\times 400 } } \)
\(=\frac { 6.626\times 10^{ -34 } }{ 4\times 20\times { 10 }^{ -23 }\sqrt { 1.67\times 1.6 } } \) = 0.00507 \(\mathring { A }\)

 \(\lambda_{p}=\frac{h}{\sqrt{2 m e V}} ; \lambda_{\alpha}=\frac{h}{\sqrt{m e V}} ; V=512 \mathrm{~V} \)
\(\frac{\lambda_{p}}{\lambda_{\alpha}}=\sqrt{\left(\frac{m_{\alpha}}{m_{p}}\right)\left(\frac{e_{\alpha}}{e_{p}}\right)\left(\frac{v_{\alpha }}{v_{p}}\right)} \)
\(\frac{m_{\propto}}{m_{p}}=4 ; \frac{e_{\alpha}}{e_{p}}=2 ; \frac{v_{\alpha}}{v_{p}}=\frac{x}{512} ; \frac{\lambda_{p}}{\lambda_{\alpha}}=1 \)
\(1=\sqrt{4 \times 2 \times\left(\frac{x}{512}\right)}=\frac{x}{64} \Rightarrow x=64 V \)

\(\text { K.E }=81.9 \times 10^{-15} \mathrm{~J} \)
\(\lambda =\frac{\mathrm{h}}{\sqrt{2 \mathrm{mk}}}=\frac{6.626 \times 10^{-34}}{\sqrt{2 \times 9.1 \times 10^{-3} \times 1836 \times 81.9 \times 10^{-15}}} \)
\(\lambda =\mathbf{4 . 0 0} \times 10^{-14} \mathrm{~m} \)