Atomic Physics

Step 1: Energy of the second excited state.
Second excited state corresponds to . Total energy is
Step 2: Relation between kinetic and potential energy.
For an electron in a hydrogen atom,
Step 3: Calculating kinetic and potential energy.

Step 4: Conclusion.
The kinetic energy is and the potential energy is .

Step 1: Total resistance of the circuit.

Step 2: Current in the circuit.

Step 3: Potential drop across potentiometer wire.

Step 4: Potential gradient.

Step 5: Calculating e.m.f. of the cell.

Step 6: Conclusion.
The e.m.f. of the cell is .

Step 1: Angle of inclination with horizontal.
Given plane makes with vertical, hence angle with horizontal is

Step 2: Acceleration of a rolling disc.
For a solid disc rolling without slipping:

Step 3: Moment of inertia of disc.

Step 4: Substituting values.

Step 5: Conclusion.
The linear acceleration of the disc is .

Step 1: Write the formula for time period of a spring-mass system.
The time period of oscillation is given by:
where is the mass and is the spring constant.

Step 2: New mass after adding another mass.
When another mass is suspended, the total mass becomes:

Step 3: Write the new time period.

Step 4: Express the new period in terms of .

Step 5: Conclusion.
The new time period of oscillation is .

Step 1: Using the Rydberg Formula.
The wavelength of emitted radiation when an electron transitions between two orbits in a hydrogen atom is given by the Rydberg formula: For transition from th orbit to the first orbit, we can rewrite the equation: Thus, solving for , we get: Step 2: Conclusion.
Thus, the value of is .

Step 1: Diffraction at a single slit.
In a diffraction experiment involving a single slit, the angular width of the central maxima depends on the wavelength of light, the distance from the slit to the screen, and the width of the slit. The relationship is given by the formula: where is the wavelength of light and is the width of the slit.
Step 2: Conclusion.
Thus, the angular width depends on the ratio of the wavelength to the slit width, but it does not directly depend on the wavelength of the light used.
Step 3: Conclusion.
Thus, the correct answer is (A) Wavelength of light used.

Step 1: Understanding the bridge formula.
In a meter bridge, the resistance ratio is given by: where and are the resistances of the two wires, is the length of the bridge where the null point is obtained, and is the total length of the bridge.

Step 2: Calculating the specific resistance ratio.
We are given that the diameters of the wires are in the ratio 3:1, and the length at the null point is 40 cm. Using the formula for resistance, , where is the specific resistance and is the cross-sectional area, we find the ratio of specific resistance as:
Step 3: Conclusion.
The correct answer is (A) 8:1.

Step 1: Understanding the work done in blowing a soap bubble.
The work done to blow a soap bubble is proportional to the surface area of the bubble. The surface area of a bubble is given by , where is the radius of the bubble.
Step 2: Surface area comparison.
For a bubble of radius , the surface area is . For a bubble of radius , the surface area is .
Step 3: Work comparison.
Since work is proportional to surface area, the work done in blowing the bubble with radius is:
Step 4: Conclusion.
The work done in blowing a bubble of radius is , so the correct answer is (C) .

Step 1: Understanding the series limits.
The series limit for the Balmer series corresponds to the wavelength when the electron jumps from the level to . The series limit for the Brackett series corresponds to the transition from to .
Step 2: Using the relationship for wavelengths.
The relation between the wavelengths for the two series is given by: Step 3: Conclusion.
The correct answer is (B), .

Step 1: Understanding the energy and angular momentum relation.
In Bohr's model of the atom, the angular momentum of an electron is quantized and given by , where is the principal quantum number. The total energy of the electron is given by .
Step 2: Deriving the relationship.
We know that and . Thus, we can derive the relation between and as: Step 3: Conclusion.
The correct answer is (C), .

Step 1: Relation of de-Broglie wavelength in Bohr orbit.
For an electron in the Bohr orbit, de-Broglie wavelength is given by:


Step 2: Relation between radius and principal quantum number.
Hence,


Step 3: Calculate ratio.
But considering total energy relation , the effective wavelength ratio becomes:


Step 4: Conclusion.
The correct ratio is .

Step 1: Understanding the relationship.
In the hydrogen atom, the angular momentum of the electron is quantized and is given by: where is the reduced Planck constant, and is the principal quantum number. The linear momentum of the electron in the hydrogen atom is related to its velocity and radius by: where is the mass of the electron, and is its velocity. Step 2: Product of angular and linear momentum.
The product of angular momentum and linear momentum is: The product is proportional to , meaning it is independent of the principal quantum number . Therefore, the correct answer is . Step 3: Conclusion.
Thus, the product of the angular and linear momentum is proportional to , which corresponds to option (B).

Step 1: Expression for magnetic field.
For the magnetic field at a point in hydrogen atom due to circular motion of the electron: This is derived from the general expression for magnetic field due to circular current in an atom.
Step 2: Conclusion.
The correct expression is option (A).

Step 1: Understanding the problem.
The electron in the hydrogen atom transitions between different energy levels. The wavelengths emitted during these transitions are related to the energy differences between the levels. The energy of a photon emitted when the electron transitions between energy levels is given by: where is the principal quantum number. The wavelength of the emitted photon is related to the energy by: where is Planck’s constant, and is the speed of light.
Step 2: Applying the energy level formula.
The transition from the third to the second excited state (i.e., to ) gives the first wavelength , and the transition from the second to the first excited state (i.e., to ) gives the second wavelength . The ratio of the wavelengths is given by: By calculating the energy differences between the two transitions, we get:
Step 3: Conclusion.
The ratio of the wavelengths is , which corresponds to option (D).

Step 1: Use Rydberg formula.


Step 2: Transition from 3 to 2.


Step 3: Transition from 4 to 3.


Step 4: Take ratio.

Step 1: Recall radius of Bohr orbit.
In hydrogen atom, radius of orbit is given by: Step 2: Write expression for area of orbit.
Area of orbit: Step 3: Identify excited states.
First excited state:
Second excited state:
Step 4: Find ratio of areas.

Step 1: Understanding energy transitions in hydrogen-like atoms.
When an electron makes a transition from an excited state to the ground state, it moves to a lower energy level. The kinetic energy (K.E.) of the electron increases because the electron is closer to the nucleus in the ground state. The potential energy (P.E.) decreases as the electron moves to a more stable state, and the total energy (T.E.) also decreases.
Step 2: Analyzing the energy changes.
In a hydrogen-like atom, the kinetic and potential energies are related, and when the electron transitions to the ground state, the total energy decreases. The correct statement is that the kinetic energy increases, but both the potential energy and the total energy decrease.
Step 3: Conclusion.
The correct answer is (C) K.E. increases but P.E. and T.E. decrease.

Step 1: Write expressions for energies.
For electron in Bohr orbit:
Step 2: Use centripetal force condition.

Step 3: Find ratio.

Step 4: Conclusion.
The ratio is .

Step 1: Formula for centripetal acceleration.
The centripetal acceleration for an electron in the -th orbit of a hydrogen atom is given by: where is the speed of the electron, is the radius of the orbit, and is the Coulomb's constant. The radius of the -th orbit is proportional to , i.e., . Step 2: Calculating the ratio.
Thus, the ratio of the centripetal accelerations for the 3rd and 5th orbits is: Step 3: Conclusion.
Thus, the correct answer is (B) .

Concept:
Last line corresponds to : Step 1: Paschen series.
Step 2: Balmer series.
Step 3: Ratio.
Step 4: Conclusion.

Step 1: Concept
The fundamental frequency of an open organ pipe is .
Step 2: Analysis
Frequency of first pipe ( ) .
Frequency of second pipe ( ) .
Step 3: Calculation
Beat frequency ( )
.
Step 4: Conclusion
Hence, the correct expression is .
Final Answer: (D)

Step 1: Formula
Work done .
Negative work ( ) implies expansion, meaning .
Step 2: Check
- (A) (Expansion, is negative). - (B) (Compression, is positive). - (C) ( ). - (D) (Compression, is positive).
Final Answer: (A)

Concept:
For vertical spring:

• Mean position is where spring is stretched by
• If spring is unstretched at highest point → amplitude

Maximum speed: Step 1: Find angular frequency.
Step 2: Find amplitude.
Step 3: Find maximum speed. Correction using correct amplitude relation (twice extension):
Step 4: Conclusion.