Network Theory

Step 1: Understanding the given network.
The network consists of linear resistors and ideal voltage sources. Ideal voltage sources maintain a fixed voltage regardless of the current drawn from them.
Step 2: Effect of doubling resistances.
When all resistances in the circuit are doubled, the total resistance of the circuit increases. As a result, the current flowing in the circuit decreases according to Ohm’s law.
Step 3: Voltage across each resistor.
Although the current decreases, the voltage distribution in a linear resistive network with ideal voltage sources depends only on the ratio of resistances, not their absolute values. Since all resistances are scaled by the same factor, their ratios remain unchanged.
Step 4: Analyzing the options.
(A) Voltage is not halved because voltage division ratios remain the same.
(B) Voltage does not double since source voltages are unchanged.
(C) Voltage cannot increase four times without change in source voltage.
(D) Correct — the voltage across each resistor remains unchanged.
Step 5: Final conclusion.
Thus, even after doubling all resistances, the voltage across each resistor remains the same.

Step 1: Energy supplied by the DC source.
When a DC voltage source of voltage is connected to a series R–C circuit, the total energy supplied by the source during charging is given by:

Step 2: Energy stored in the capacitor at steady state.
At steady state, the capacitor is fully charged and the energy stored in it is:

Step 3: Energy dissipated in the resistor.
The remaining energy supplied by the source is dissipated as heat in the resistor. Hence, half of the supplied energy is lost in resistance.
Step 4: Ratio calculation.

Step 5: Final conclusion.
Therefore, the ratio of energy stored in the capacitor to the total energy supplied by the source is 0.5.

Step 1: Understanding nodal analysis.
Nodal analysis is a systematic method used to determine the voltages at different nodes of an electrical circuit with respect to a reference node (usually ground). The method focuses on node voltages rather than loop currents.
Step 2: Identifying the governing law.
At each node (except the reference node), Kirchhoff’s Current Law (KCL) is applied. According to KCL, the algebraic sum of currents entering and leaving a node is zero.
Step 3: Use of Ohm’s law.
To express currents in terms of node voltages, Ohm’s law is used. Currents through resistive elements are written as the voltage difference divided by resistance.
Step 4: Eliminating incorrect options.
(A) KVL is not directly used in nodal analysis.
(C) KVL applies mainly to mesh analysis.
(D) Although Ohm’s law is used, KVL is not a fundamental requirement of nodal analysis.
Step 5: Final conclusion.
Hence, nodal analysis is fundamentally based on KCL and Ohm’s law.