When connected to a d.c. supply, the voltage across a coil measured by a voltmeter is \(1 \mathrm{~V}\) when the current through it measured by an ammeter is \(1 \mathrm{~mA}\). The same coil across a \(50 \mathrm{~Hz}\) a.c. supply gives a.c. meter readings of \(10 \mathrm{~V}\) and \(4 \mathrm{~mA}\). What are the resistance and inductance of the coil, assuming that the meters are ideal?

Understanding Ohm's Law serves as the foundation for analyzing electric circuits. It explains the relationship between voltage (V), current (I), and resistance (R) in an electrical circuit. The law is expressed by the equation \( V = IR \). This signifies that the voltage across a conductor is directly proportional to the current flowing through it, provided the temperature and other physical conditions remain constant.

To apply Ohm's Law to a real-world scenario, consider a coil connected to a direct current (d.c.) source. The resistance can be calculated by rearranging Ohm's Law to \( R = \frac{V}{I} \) and using the voltage and current measurements. For instance, a coil with a 1 Volt direct current through it and a current of 1 milliampere exhibits a resistance of 1000 ohms, as seen in our exercise.

Inductive reactance (\(X_L\)) is a concept specifically related to alternating current (a.c.) circuits. Unlike resistance, which hinders both a.c. and d.c. current, inductive reactance only opposes the change in current found in a.c. systems. It emerges due to the inductance in a coil, causing a phase shift between the voltage and current. The formula for calculating inductive reactance is \( X_L = 2\pi fL \) where \( f \) is the frequency of the a.c. supply and \( L \) is the inductance.

Calculating inductive reactance requires the known values of frequency and inductance. For instance, if a coil placed in a 50 Hz a.c. circuit shows a reactance of 2000 ohms, it tells us that the coil is hindering the alternating current’s phase-shifting ability to a specific extent.

Impedance \(Z\) is a comprehensive measure of obstruction that a circuit presents to the flow of alternating current (a.c.). It is akin to resistance in a direct current (d.c.) circuit but includes both resistive and reactive (inductive or capacitive) resistance. The formula to express impedance is \( Z = \sqrt{R^2 + X^2} \), with \( R \) representing resistance and \( X \) being the reactance, which can either be inductive (\(X_L\)) or capacitive (\(X_C\)) depending on the type of circuit.

In the coil example, the impedance calculation reveals that the total obstruction to the a.c. is more significant than just the resistance portion. This higher impedance value accounts for both the resistance of the wire and the inductive reactance, illustrating the coil’s behavior in an a.c. circuit.

Electric circuits can be categorized based on the type of current they carry: direct current (d.c.) or alternating current (a.c.). In a direct current circuit, electricity flows steadily in one direction. In contrast, in an alternating current circuit, electricity changes direction periodically.

Differences in these types of circuits lead to different behaviors of components such as coils. In d.c. circuits, a coil resists current due to its resistance. Conversely, in a.c. circuits, a coil resists not only due to its resistance but also because of its inductive reactance, contributing to a phase shift between current and voltage. This characteristic results in a different measure for the total opposition to current flow, known as impedance, which is usually more significant than the simple resistance in d.c. circuits.