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In an accelerator experiment on high-energy collisions of electrons with positrons, a certain event is interpreted as annihilation of an electron-positron pair of total energy 10.2 BeV into two $\gamma$-rays of equal energy. What is the wavelength associated with each $\gamma$-ray? $\left(1 \mathrm{BeV}=10^9 \mathrm{eV}\right)$
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Given: Energy of $2 \gamma$-rays $=10.2 \mathrm{BeV}$
Energy of each $\gamma$-ray $=\frac{10.2}{2}=5.1 \mathrm{BeV}$ $=5.1 \times 10^9 \mathrm{eV}=5.1 \times 10^9 \times 1.6 \times 10^{-19} \mathrm{~J}$
Now,
$$
\begin{aligned}
\lambda &=\frac{h c}{\mathrm{E}}=\frac{6.6 \times 10^{-34} \times 3 \times 10^8}{5.1 \times 1.6 \times 10^{-10}} \\
&=2.434 \times 10^{-16} \mathrm{~m}
\end{aligned}
$$
Energy of each $\gamma$-ray $=\frac{10.2}{2}=5.1 \mathrm{BeV}$ $=5.1 \times 10^9 \mathrm{eV}=5.1 \times 10^9 \times 1.6 \times 10^{-19} \mathrm{~J}$
Now,
$$
\begin{aligned}
\lambda &=\frac{h c}{\mathrm{E}}=\frac{6.6 \times 10^{-34} \times 3 \times 10^8}{5.1 \times 1.6 \times 10^{-10}} \\
&=2.434 \times 10^{-16} \mathrm{~m}
\end{aligned}
$$
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