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An alpha particle travels in a circular path of radius $0.45$m in a magnetic field with $B=1.2$ w/m$^2$. Calculate (i) its speed (ii) its period of revolution, and (iii) its kinetic energy. Mass of proton particle = \(1.67\times 10^{-27}\)kg \(\approx 4\times M_p= 4\times938.27\) MeV.

Solution :

• [(i)] the magnetic force \(eBv\) must be equal to the mass times acceleration. Therefore \begin{equation*} Bev = \frac{Mv^2}{R}, \end{equation*} where \(R\) is the radius of the circular orbit. Hence \begin{equation*} v= \frac{eBR}{M} = \frac{2\times1.6 \times10^{-19}\times 1.2 \times 0.45}{4\times 1.67\times 10^{-27}}\approx 2.7 \times10^7 \text{m/s}. \end{equation*}
• [(ii)] The time period is \begin{equation*} T = \frac{2\pi R}{v} = \frac{2\times3.14\times 0.45}{2.7\times10^7} \approx10^{-7} \text{ s}. \end{equation*}
• [(iii)] The kinetic energy is given by \begin{eqnarray}\nonumber \text{K.E.} &=& \frac{1}{2} M v^2= \frac{1}{2}\times (4\times 1.67 \times 10^{-27}) \times \big(2.7\times10^7\big)^2 \\\nonumber &=& 3.26\times 7.29 \times 10^{-13} \approx 23.7 \times 10^{-13} \text{J}. \end{eqnarray}

Consider a closed cylinder whose walls are adiabatic. The cylinder is divided into three equal parts $A_1$, $A_2$ and $A_3$ by means of partitions $S_1$ and $S_2$, which can move along the length of the cylinder without friction. The partition $S_1$ is adiabatic and $S_2$ is conducting. Initially, each of the three parts contain one mole of Helium gas, which can be treated as an ideal gas, is at pressure $P_0$, temperature $T_0$ and volume $V_0$. Assume the specific heat at constant volume $C_v\,=\,\frac{3R}{2}$ and the specific heat at constant pressure $C_p\,=\,\frac{5R}{2}$. Now, heat is supplied to the to the left most partition $A_1$ till the temperature in part $A_3$ becomes $T_3\,=\,\frac{9T_0}{4}$ Find the final volume, pressure and temperature in terms of $V_0$, $P_0$ and $T_0$. Assume the entire process is quasistatic.
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3. ( Continuation of problem 2)

(a) What is work done by the gas in $A_1$ ?

(b) What is the heat supplied to the gas in $A_1$?

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