Communication Systems

We know from the first law of thermodynamics:

Where:
- is the heat supplied to the gas,
- is the change in internal energy,
- is the work done by the gas.
The work done is: The change in internal energy for a diatomic gas is: The heat supplied is: Combining these expressions, we get: Simplifying: Solving for the molar heat capacity: Thus, the molar heat capacity is:

The wavelengths of the Balmer series are given by the Balmer formula: where is the Rydberg constant, is the wavelength, and is the principal quantum number, with .
For the first Balmer line (H , ): For the second Balmer line (H , ): The ratio of these wavelengths:

We are given the inductance , the resistance , and the voltage .
To find the energy stored in the magnetic field, we use the formula: First, calculate the current using Ohm’s law: Now, substitute the values into the energy formula: Thus, the energy stored in the magnetic field is .

The first law of thermodynamics states:


Where:
- is the heat supplied to the system,
- is the change in internal energy,
- is the work done by the system.
The first law introduces the concept of internal energy and explains how energy is conserved in the system. It is essentially a restatement of the law of conservation of energy.

However, the concept of entropy is not introduced by the first law but by the second law of thermodynamics, which deals with the direction of processes and the measure of disorder in the system. Therefore, the statement in option (2) is incorrect.

Using Bernoulli's principle:
Where:
- (initial pressure),
- (final pressure),
- , - (since the pipe is closed initially),
- is the speed of water we need to find.
Rearranging the equation for : Substitute the known values: Thus, the speed of water flowing out of the pipe is .

The car is moving on a circular path, so it experiences both tangential and centripetal acceleration.
Step 1: Centripetal acceleration The centripetal acceleration is given by the formula: where and . Substituting the values:
Step 2: Total acceleration The total acceleration is the sum of the tangential and centripetal accelerations: where is the tangential acceleration. Substituting the values: Thus, the total acceleration is .

Using the decay formula where is the remaining amount, is the initial amount, is the decay constant, and is time.
We set up equations for the 44% and 93% decay points: For 44% decay (56% remaining): For 93% decay (7% remaining): The time gap between these two decays is given as 81 minutes: The half-life of the substance is given by : Calculating gives us the half-life:

The modulation index is given by the ratio of the peak voltage of the modulating signal to the peak voltage of the carrier wave : Given that and , we can solve for : Thus, the peak voltage of the modulating signal is 12 V.
Final Answer: 12 V.

When the metal plate is inserted, the capacitance of the capacitor changes. Let be the initial capacitance and be the final capacitance.
The initial capacitance is After the metal plate is inserted, the capacitor is effectively divided into two capacitors in series, each with plate separation . The capacitance of each of these capacitors is so the final capacitance is which means . The initial energy stored in the capacitor is Since the battery is disconnected, the charge on the capacitor remains constant. Then , where is the final potential difference across the capacitor. Hence, The final energy stored in the capacitor is The work done on the metal slab is equal to the change in energy stored in the capacitor: Final Answer: