Answer the following questions:
$(a)$ In any $ac$ circuit, is the applied instantaneous voltage equal to the algebraic sum of the instantaneous voltages across the series elements of the circuit? Is the same true for $rms$ voltage?
$(b)$ Why is a capacitor used in the primary circuit of an induction coil?
$(c)$ An applied voltage signal consists of a superposition of a $dc$ voltage and an $ac$ voltage of high frequency. The circuit consists of an inductor and a capacitor in series. Show that the $dc$ signal will appear across $C$ and the $ac$ signal across $L$.
$(d)$ $A$ choke coil in series with a lamp is connected to a $dc$ line. The lamp is seen to shine brightly. Insertion of an iron core in the choke causes no change in the lamp's brightness. Predict the corresponding observations if the connection is to an $ac$ line.
$(e)$ Why is a choke coil needed in the use of fluorescent tubes with $ac$ mains? Why can we not use an ordinary resistor instead of the choke coil?

(N/A) Yes, the applied instantaneous voltage is equal to the algebraic sum of the instantaneous voltages across the series elements of the circuit due to Kirchhoff's Voltage Law. However, this is not true for $rms$ voltage because the voltages across different elements are generally not in phase.
$(b)$ $A$ capacitor is used in the primary circuit of an induction coil to prevent sparking at the contact breaker. When the circuit is broken, the rapid change in current induces a high back $emf$ in the primary coil, which would cause a spark across the contacts. The capacitor provides a path for the current, absorbing the energy and preventing the spark.
$(c)$ For a $dc$ signal, the frequency is $0$. The inductive reactance $X_L = 2\pi fL = 0$, while the capacitive reactance $X_C = 1/(2\pi fC) \to \infty$. Thus, the $dc$ voltage drops across $C$. For a high-frequency $ac$ signal, $X_L$ is very large and $X_C$ is very small. Thus, the $ac$ voltage drops across $L$.
$(d)$ If connected to an $ac$ line, the lamp will glow less brightly when an iron core is inserted. The iron core increases the inductance $L$ of the choke, which increases the inductive reactance $X_L = 2\pi fL$. This increases the total impedance of the circuit, reducing the current and thus the brightness of the lamp.
$(e)$ $A$ choke coil is used to limit the current in a fluorescent tube circuit without dissipating significant power, as it has a high reactance and low resistance. An ordinary resistor would dissipate power as heat $(I^2R)$, which is inefficient and wasteful.