Atomic Physics

The Stoke's line wavenumber change is related to the rotational energy levels. The spacing between consecutive S-branch lines is given as 243.2 cm . For the next energy level, the wavenumber of the Stoke's line will increase by the same amount. So, the wavenumber for the next higher energy level is: Thus, the wavenumber for the next higher energy level is .

In the absence of an external magnetic field, the transition between the and states would result in a single transition line. However, in the presence of a weak magnetic field, the degeneracy of the energy levels is lifted, leading to the splitting of the transition line. For the state, the total angular momentum , and for the state, . The magnetic field causes the splitting of these levels according to the Zeeman effect. The total number of possible lines that can appear is determined by the number of different transitions between the values of the and states. For , there are 4 possible values ( ), and for , there are 2 possible values ( ). Therefore, the total number of lines that can appear is: Thus, the number of lines that will appear in the presence of a weak magnetic field is 6.

Step 1: The effective nuclear charge is , so the atomic number can be assumed to be 2.
For the first ionization, the electron is removed from the energy level, so the energy of the first ionization is: The ionization potential is the absolute value of the energy, so the first ionization potential is: Step 2: The ionization potential is approximately atomic units.

Let's analyze each option:

1. Linear Stark effect is zero:
In the ground state of a hydrogen atom (which is the state), the electric dipole moment is zero because the electron is spherically symmetric. Hence, the linear Stark effect, which is proportional to the electric dipole moment, is zero in this case.
Therefore, (A) is correct.

2. It has definite parity:
The parity of a quantum state is determined by the wavefunction. The wavefunction for the hydrogen atom in the ground state ( state) is spherically symmetric, and therefore, the parity of the ground state is well-defined.
Therefore, (B) is correct.

3. Spin-orbit coupling is zero:
The ground state of the hydrogen atom is in the orbital, which has no orbital angular momentum ( ). Since spin-orbit coupling is proportional to the interaction between the electron’s spin and its orbital angular momentum, and in the ground state, the spin-orbit coupling is zero.
Therefore, (C) is correct.

4. Hyperfine splitting is zero:
Hyperfine splitting arises due to the interaction between the nuclear spin and the electronic spin. In the case of the hydrogen atom, this interaction is not exactly zero, but it is extremely small. However, it is not exactly zero, so this statement is incorrect.
Therefore, (D) is incorrect.

Step 1: Understanding the Concept:
Faraday's Law states that a time-varying magnetic field induces an electromotive force (EMF), which in turn creates a circulating electric field.
Step 2: Key Formula or Approach:
The integral form is .
Using Stokes' Theorem, this is converted into the differential form used in Maxwell's equations.
Step 3: Detailed Explanation:
Let's look at the options:
- (A) is Gauss's Law for Electricity.
- (B) is Gauss's Law for Magnetism (no monopoles).
- (C) relates the curl of the electric field to the time rate of change of the magnetic field. This is the differential form of Faraday's Law.
- (D) (ignoring displacement current) is Ampere's Circuital Law.
Step 4: Final Answer:
The equation represents Faraday’s law.