Question

The nucleus of 8Be, which consists of 4 protons and 4 neutrons, is very unstable and spontaneously breaks into two alpha particles(helium nuclei, each consisting of 2 protons and 2 neutrons).

(a)What is the force between the two alpha particles when they are5.00 X 10^-15m apart, and (b) what will be the magnitude of the acceleration of the alpha particles due to this force? Note that the mass of an alpha particle is 4.0026 u.

Answer 1

Concepts and reason

The concept required to solve this problem is the coulombs law. Initially, calculate the force between two alpha particles by using the coulombs law. Next, write the expression for the acceleration by using Newton's second law of motion. Finally, calculate the acceleration of the alpha particle by using the expression for the acceleration.

Fundamentals

According to Coulomb's law, the force between any two charged particles is proportional to the product of two charges and inversely proportional to the square of the distance between the two charged particles. The formula for Coulomb force between two charged particles is, \(F=\frac{1}{4 \pi \varepsilon 0} \frac{q 1 g 2}{d^{2}}\)

Here, \(q_{1}, q_{2}\) are two charges, \(\varepsilon_{0}\) is the permittivity of the free space, and \(d\) is the distance between the two charges. Charges are of two types one is a positive charge, and another one is a negative charge. When a charge is at rest position, it produces only an electric field. Positive charges diverge the electric field lines, and negative charges converge the electric field lines. If the two charges are unlike charges then the force between them is attractive coulombic force, and if the two charges are like charges, then the force between them is repulsive coulombic force.

(a) Alpha particle has two positive charge particles, protons, and two neutrons, but no electrons. This is why alpha particle is also called helium nucleus because helium nucleus also has two protons and two neutrons. The charge of the alpha particle is, \(q=2 e\)

Here, e is the charge of the proton. Substitute \(1.6 \times 10^{-19} C\) for e.

$$ \begin{aligned} q=& 2\left(1.6 \times 10^{-19} C\right) \\ &=3.2 \times 10^{-19} C \end{aligned} $$

The coulombic force between the two alpha particles having the same charge is, \(F=\frac{1}{4 \pi \varepsilon 0} \frac{q^{2}}{r^{2}}\)

Substitute \(3.2 \times 10^{-19} C\) for \(q\) and \(5 \times 10^{-15} m\) for \(r\) and \(8.854 \times 10^{-12} C^{2} / N m^{2}\) for \(\varepsilon_{0}\) in the above equation.

$$ \begin{array}{c} F=\frac{1}{4 \pi\left(8.854 \times 10^{-12} C^{2} / N m^{2}\right)} \frac{\left(3.2 \times 10^{-19} C\right)^{2}}{\left(5 \times 10^{-15} m\right)^{2}} \\ =\left(9 \times 10^{9} N m^{2} / C^{2}\right) \frac{10.24 \times 10^{-38} C^{2}}{25 \times 10^{-30} m^{2}} \\ =36.86 N \end{array} $$

Therefore, the force between the two alpha particles is \(36.86 N\).

Alpha particle has two positive charge particles means protons and two neutrons, but no electrons. This is why alpha particle is also called helium nucleus because helium nucleus also has two protons and two neutrons. Both the charges have the same charge. So, the force between the two alpha particles is repulsive. Alpha particle has two protons. So, the total charge of the alpha particle is adding the magnitude of two charges.

(b) The formula for the force acting on the alpha particle, which is moving acceleration is, \(F=m a\)

Here, \(\mathrm{m}\) is the alpha particle's mass, and a is the acceleration of the alpha particle. Rearrange the above equation for acceleration.

$$ a=\frac{F}{m} $$

Substitute \(4.0026 \mathrm{u}\) for \(\mathrm{m}\) and \(36.86 N\) for \(\mathrm{F}\) in the above equation.

$$ \begin{aligned} a &=\frac{36.86 N}{4.0026 u\left(\frac{1.67 \times 10^{-27} k g}{1 u}\right)} \\ &=5.514 \times 10^{27} \mathrm{~m} / \mathrm{s}^{2} \end{aligned} $$

Therefore, the acceleration of the alpha particle is \(5.514 \times 10^{27} \mathrm{~m} / \mathrm{s}^{2}\)

According to Newton's second law, the force \(F\) is equal to the product of the mass of the alpha particle and acceleration of the alpha particle. Acceleration is the rate of change in velocity.

Part a The force between the two alpha particles is \(36.86 N\)

Part \(b\) The acceleration of the alpha particle is \(5.514 \times 10^{27} \mathrm{~m} / \mathrm{s}^{2}\)