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University Physics Volume 3

17th Edition
William Moebs + 1 other
ISBN: 9781938168185
BuyFindarrow_forward

University Physics Volume 3

17th Edition
William Moebs + 1 other
ISBN: 9781938168185
Textbook Problem

If two nuclei are to fuse in a nuclear reaction, they must be moving fast enough so that the repulsive Coulomb force between them does not prevent them for getting within R 10 14 m of one another. At this distance or nearer, the attractive nuclear force can overcome the Coulomb force, and the nuclei are able to fuse.

(a) Find a simple formula that can be used to estimate the minimum kinetic energy the nuclei must have if they are to fuse. To keep the calculation simple, assume the two nuclei are identical and moving toward one another with the same speed v. (b) Use this minimum kinetic energy to estimate the minimum temperature a gas of the nuclei must have before a significant number of them will undergo fusion. Calculate this minimum temperature first for hydrogen and then for helium. (Hint: For fusion to occur, the minimum kinetic energy when the nuclei are far apart must be equal to the Coulomb potential energy when they are a distance R apart.)

To determine

(a)

The minimum kinetic energy of two identical nuclei moving together.

Explanation

Given:

The distance between two nucleuses is, R=1014m.

Formula used:

The kinetic energy is given by,

  K.E=12mv2

Here, the mass of nuclei is m and the velocity if nuclei is v.

The total kinetic energy of two nuclei moving same direction and having same velocity is given by,

  (K.E)T=(K.E)1+(K.E)2

Here, the total kinetic energy is (K.E)T, the kinetic energy of first nuclei is (K.E)1 and the kinetic energy of second nuclei is (K.E)2.

Hence, two nuclei moving in same direction and having same velocity, (K.E)1=(K.E)2.

Solve further,

  (K.E)T=2(K.E)

The coulombs force of two same charges is given by,

  Fcoulomb=14πε0(q2R2)

Here, the charge of nucleus is q and the permittivity of free space is ε0

To determine

(b)

The minimum temperature of Hydrogen and the minimum temperature of Helium.

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