To calculate the initial magnitude and direction of the acceleration of the electron, let's go through the following steps:
1. Given Values:
- The initial distance between the electron and the carbon atom: [tex]\( r = 4.3 \times 10^{-6} \)[/tex] meters.
- The polarizability of the carbon atom: [tex]\( \alpha = 1.96 \times 10^{-40} \, \text{C} \cdot \text{m} / (\text{N} / \text{C}) \)[/tex].
- The charge of an electron: [tex]\( q_e = -1.602 \times 10^{-19} \)[/tex] Coulombs.
- The mass of an electron: [tex]\( m_e = 9.109 \times 10^{-31} \)[/tex] kg.
- Coulomb's constant: [tex]\( k_e = 8.99 \times 10^9 \, \text{N} \cdot \text{m}^2 / \text{C}^2 \)[/tex].
2. Calculate the Force:
- The force between the electron and the induced dipole on the carbon atom can be given by Coulomb's law:
[tex]\[
F = k_e \frac{q_e^2}{r^2}
\][/tex]
3. Calculate the Numerical Value of the Force:
- Use the given values and plug them into the equation:
[tex]\[
F = \left( 8.99 \times 10^9 \, \text{N} \cdot \text{m}^2 / \text{C}^2 \right) \cdot \frac{\left(-1.602 \times 10^{-19}\, \text{C} \right)^2}{\left( 4.3 \times 10^{-6}\, \text{m} \right)^2}
\][/tex]
- After evaluating the expression:
[tex]\[
F \approx 1.247808110329908 \times 10^{-17} \, \text{N}
\][/tex]
4. Calculate the Acceleration of the Electron:
- Use Newton's second law to find the acceleration:
[tex]\[
a = \frac{F}{m_e}
\][/tex]
- Plug in the values:
[tex]\[
a = \frac{1.247808110329908 \times 10^{-17} \, \text{N}}{9.109 \times 10^{-31} \, \text{kg}}
\][/tex]
- After evaluating the expression:
[tex]\[
a \approx 1.369862894203434 \times 10^{13} \, \text{m/s}^2
\][/tex]
5. Direction of Acceleration:
- The electron is negatively charged and will be attracted towards the positively induced side of the neutral carbon atom. Therefore, the direction of the acceleration will be towards the carbon atom.
Summary:
- The initial magnitude of the acceleration of the electron: [tex]\(1.369862894203434 \times 10^{13} \, \text{m/s}^2\)[/tex]
- The direction of the acceleration is towards the carbon atom.
