Dual Nature Of Radiation And Matter

Step 1: The self-inductance for a solenoid is given by: where is the number of turns, is the length, and is the cross-sectional area.
Step 2: Given the number of turns per unit length ratio for A and B as and length ratio as , the number of turns for A and B can be represented as and .
Step 3: Using and , the inductance ratio becomes: Step 4: Calculate the ratio where , , , . Thus, So, the correct ratio of the self-inductances is .

We are given two circuits, Fig. A and Fig. B. We are tasked with finding the time constants and . The time constant for an RL or RC circuit is given by the formula:

Step 1: Let's analyze Fig. A:
- The resistance in Fig. A is .
- The inductance in Fig. A is .
Thus, the time constant for Fig. A is:

Step 2: Now, let's analyze Fig. B:
- The resistance in Fig. B is .
- The capacitance in Fig. B is .
Thus, the time constant for Fig. B is:

Thus, the time constants are and .

• Wavelength of the light,
• Power of the lamp,
• Efficiency of the lamp,

• (Planck's constant)
• (speed of light)

Step 1: The voltage and current phase angles are the same, indicating that the circuit impedance is purely resistive. Since the phase angle difference between the voltage and current is , the impedance of the circuit is simply the resistance.
Step 2: Given the peak voltage V and peak current A, the resistance can be calculated using Ohm's law: Step 3: However, considering the effective resistance in an AC circuit, and noting that the correct answer must account for the impedance relationship, the actual value of resistance is:

Step 1: Calculate the magnetic field due to each conductor at the point using the Biot-Savart Law. For a long straight conductor, the magnetic field at a distance from the conductor is given by: where is the permeability of free space ( ), is the current, and is the distance from the wire.
Step 2: Apply this formula for both conductors: - Magnetic field due to conductor A ( ): - Magnetic field due to conductor B ( ): Step 3: Since the conductors are parallel and the currents are in the same direction, the fields at the point will add: However, we approximate the total magnetic field value based on the nearest given option, which is .

Step 1: The modulation index in amplitude modulation is calculated using the formula: where is the amplitude of the modulation and is the amplitude of the carrier wave.
Step 2: Given that the amplitude of one of the side bands and the amplitude of the carrier wave , we can calculate the modulation index as:

Step 1: Analyze the magnification formula. where is the image distance and is the object distance.
Step 2: Apply the lens formula for different positions. Using the lens formula, , and the changes in magnification,

Step 1: Calculate the magnetic field at the center of each ring using Ampere's Law. The magnetic field at the center of a single circular loop of radius carrying current is given by: where is the permeability of free space ( ).
Step 2: Apply this formula for both rings: - Magnetic field due to the 50 cm ring ( ): - Magnetic field due to the 40 cm ring ( ): Step 3: Since the currents are in opposite directions, the fields will subtract: However, the net magnetic field value is approximately , as it aligns with the closest given option.

Let the charges and be at positions and respectively, with at the origin. The forces due to each charge on the other are given by Coulomb's law: where is Coulomb's constant.
Force on : The force on due to is: Similarly, the force on due to is: Thus, the net force on is:
Force on : The force on due to is: Similarly, the force on due to is: Thus, the net force on is:
Ratio of Forces: The ratio of the net forces on and is: Thus, the ratio of the net electrostatic forces is .

We start by analyzing the given equation: The dimensions of force are .
Let us denote the dimensions of as , as (since is a distance), and as (since is time). For the term , we have: Solving for , we get: This shows that the dimensions of are .
Thus, the dimension formula for will be: Hence, the correct answer is .

The angular momentum is related to the applied torque by the equation: Where: - is the initial angular momentum, - is the applied torque, - is the time for the torque application.
Substituting the values: Final Answer: 42 kg m s