Consider an electron in a hydrogen atom, which is \(0.050 \mathrm{nm}\) from the proton in the nucleus. a) What electric field does the electron experience? b) In order to produce an electric field whose root-meansquare magnitude is the same as that of the field in part (a), what intensity must a laser light have?

To find the electric field experienced by the electron, we'll use the formula for Coulomb's law, imagining that there's an electron at \(0.050 \mathrm{nm}\) from the proton in the nucleus: \( E = \frac{k * q_1 * q_2}{r^2} \) Where: - E is the electric field, - k is the electrostatic constant (\(8.99 * 10^9 \mathrm{Nm^2C^{-2}}\)), - \(q_1\) and \(q_2\) are the charges of the proton and electron, respectively (both have the same magnitude of charge \(e\): \(1.6 * 10^{-19}\ \mathrm{C}\)), - r is the distance between the proton and electron (\(0.050\ \mathrm{nm}\)). Plugging in the values: \( E = \frac{(8.99 * 10^9 \mathrm{Nm^2C^{-2}}) * (1.6 * 10^{-19}\ \mathrm{C})^2}{(0.050 * 10^{-9}\ \mathrm{m})^2} \) After calculating, we get: \( E \approx 4.58 * 10^{11}\ \mathrm{N/C} \)

We will now find the relationship between the electric field and the intensity of a laser light. In a plane electromagnetic wave, we know that: \( S = \frac{1}{2} * {E_{rms}}^2 * \epsilon_0 * c \) Where: - S is the power per unit area (intensity), - \(E_{rms}\) is the root-mean-square electric field, - \(\epsilon_0\) is the vacuum permittivity (\(\approx 8.85 * 10^{-12} \mathrm{C^2/Nm^2}\)), - c is the speed of light in vacuum (\(\approx 3 * 10^8 \mathrm{m/s}\)). Since we want the intensity to have the same root-mean-square magnitude as the electric field calculated in part (a), we can rewrite the above equation as: \( S = \frac{1}{2} * E^2 * \epsilon_0 * c \)

Plugging in the values from step 1 for the electric field, and the values for \(\epsilon_0\) and c: \( S = \frac{1}{2} * ({4.58 * 10^{11}\ \mathrm{N/C}})^2 * (8.85 * 10^{-12}\ \mathrm{C^2/Nm^2}) * (3 * 10^8\ \mathrm{m/s}) \) Calculating this expression, we get: \( S \approx 2.83 * 10^9 \mathrm{W/m^2} \) So in order to produce an electric field with the same root-mean-square magnitude as that in the hydrogen atom, the laser light must have an intensity of approximately \(2.83 * 10^9 \mathrm{W/m^2}\).