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JEE Advance - Physics (2009 - Paper 1 Offline - No. 17)

Scientists are working hard to develop nuclear fusion reactor. Nuclei of heavy hydrogen, $$_1^2$$H, known as deuteron and denoted by D, can be thought of as a candidate for fusion reactor. The D-D reaction is $$_1^2$$H + $$_1^2$$H $$\to$$ $$_2^3$$He + $$n$$ + energy. In the core of fusion reactor, a gas of heavy hydrogen is fully ionized into deuteron nuclei and electrons. This collection of $$_1^2$$H nuclei and electrons is known as plasma. The nuclei move randomly in the reactor core and occasionally come close enough for nuclear fusion to take place. Usually, the temperatures in the reactor core are too high and no material wall can be used to confine the plasma. Special techniques are used which confine the plasma for a time $$t_0$$ before the particles fly away from the core. If $$n$$ is the density (number/volume) of deuterons, the product $$nt_0$$ is called Lawson number. In one of the criteria, a reactor is termed successful if Lawson number is greater than 5 $$\times$$ 10$$^{14}$$ s/cm$$^3$$.

It may be helpful to use the following : Boltzmann constant $$k = 8.6 \times {10^{ - 5}}$$ eV/K; $${{{e^2}} \over {4\pi {\varepsilon _0}}} = 1.44 \times {10^9}$$ eVm.

Scientists are working hard to develop nuclear fusion reactor. Nuclei of heavy hydrogen, $$_1^2$$H, known as deuteron and denoted by D, can be thought of as a candidate for fusion reactor. The D-D reaction is $$_1^2$$H + $$_1^2$$H $$\to$$ $$_2^3$$He + $$n$$ + energy. In the core of fusion reactor, a gas of heavy hydrogen is fully ionized into deuteron nuclei and electrons. This collection of $$_1^2$$H nuclei and electrons is known as plasma. The nuclei move randomly in the reactor core and occasionally come close enough for nuclear fusion to take place. Usually, the temperatures in the reactor core are too high and no material wall can be used to confine the plasma. Special techniques are used which confine the plasma for a time $$t_0$$ before the particles fly away from the core. If $$n$$ is the density (number/volume) of deuterons, the product $$nt_0$$ is called Lawson number. In one of the criteria, a reactor is termed successful if Lawson number is greater than 5 $$\times$$ 10$$^{14}$$ s/cm$$^3$$.

It may be helpful to use the following : Boltzmann constant $$k = 8.6 \times {10^{ - 5}}$$ eV/K; $${{{e^2}} \over {4\pi {\varepsilon _0}}} = 1.44 \times {10^9}$$ eVm.

Scientists are working hard to develop nuclear fusion reactor. Nuclei of heavy hydrogen, $$_1^2$$H, known as deuteron and denoted by D, can be thought of as a candidate for fusion reactor. The D-D reaction is $$_1^2$$H + $$_1^2$$H $$\to$$ $$_2^3$$He + $$n$$ + energy. In the core of fusion reactor, a gas of heavy hydrogen is fully ionized into deuteron nuclei and electrons. This collection of $$_1^2$$H nuclei and electrons is known as plasma. The nuclei move randomly in the reactor core and occasionally come close enough for nuclear fusion to take place. Usually, the temperatures in the reactor core are too high and no material wall can be used to confine the plasma. Special techniques are used which confine the plasma for a time $$t_0$$ before the particles fly away from the core. If $$n$$ is the density (number/volume) of deuterons, the product $$nt_0$$ is called Lawson number. In one of the criteria, a reactor is termed successful if Lawson number is greater than 5 $$\times$$ 10$$^{14}$$ s/cm$$^3$$.

It may be helpful to use the following : Boltzmann constant $$k = 8.6 \times {10^{ - 5}}$$ eV/K; $${{{e^2}} \over {4\pi {\varepsilon _0}}} = 1.44 \times {10^9}$$ eVm.

Assume that two deuteron nuclei in the core of fusion reactor at temperature T are moving towards each other, each with kinetic energy 1.5 kT, when the separation between them is large enough to neglect Coulomb potential energy. Also neglect any interaction from other particles in the core. The minimum temperature T required for them to reach a separation of 4 $$\times$$ 10$$^{-15}$$ m is in the range
$$1.0 \times {10^9}K < T < 2.0 < {10^9}K$$
$$2.0 \times {10^9}K < T < 3.0 < {10^9}K$$
$$3.0 \times {10^9}K < T < 4.0 < {10^9}K$$
$$4.0 \times {10^9}K < T < 5.0 < {10^9}K$$

Explanation

To determine the minimum temperature required for two deuteron nuclei to reach a separation of $$4 \times 10^{-15}$$ m, we need to equate the initial kinetic energy of the deuterons with the potential energy at the specified separation distance.

The initial kinetic energy of each deuteron is given as :

$$ KE = 1.5 \, k \, T $$

Since there are two deuterons, the total kinetic energy is :

$$ KE_{\text{total}} = 2 \times 1.5 \, k \, T = 3 \, k \, T $$

The Coulomb potential energy (PE) between two deuterons at a separation distance $$r = 4 \times 10^{-15}$$ m is given by :

$$ PE = \frac{e^2}{4 \pi \epsilon_0 r} $$

Using the given value:

$$ \frac{e^2}{4 \pi \epsilon_0} = 1.44 \times 10^9 \, \text{eV} \cdot \text{m} $$

So, the potential energy becomes :

$$ PE = \frac{1.44 \times 10^9 \, \text{eV} \cdot \text{m}}{4 \times 10^{-15} \, \text{m}} $$

Simplifying :

$$ PE = \frac{1.44 \times 10^9}{4 \times 10^{-15}} \, \text{eV} $$

$$ PE = 0.36 \times 10^{24} \, \text{eV} $$

Now, we equate the total kinetic energy to the potential energy to find the temperature $ T $ :

$$ 3 \, k \, T = 0.36 \times 10^{24} \, \text{eV} $$

Solving for $ T $ :

$$ T = \frac{0.36 \times 10^{24}}{3 \times 8.6 \times 10^{-5}} \, \text{K} $$

$$ T = \frac{0.36 \times 10^{24}}{2.58 \times 10^{-4}} \, \text{K} $$

$$ T = 1.395 \times 10^{9} \, \text{K} $$

This temperature is in the range :

Option A: $$1.0 \times {10^9}K < T < 2.0 < {10^9}K$$

So, the correct answer is :

Option A : $$1.0 \times {10^9}K < T < 2.0 < {10^9}K$$

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