Electrical Properties of Materials and Circuit Analysis

Electrical Resistivity of Materials

Materials are classified as conductors, semiconductors, and insulators depending on their resistivities, in an increasing order of their values.

• Conductors (e.g., Metals): Have low resistivities, typically in the range of 10^–8 Ωm to 10^–6 Ωm.
• Insulators (e.g., Ceramic, Rubber, Plastics): Have resistivities 10¹⁸ times greater than metals or more.
• Semiconductors: Fall in between conductors and insulators. Their resistivities characteristically decrease with a rise in temperature. The presence of small amounts of impurities also affects their resistivities. This last feature is exploited in the use of semiconductors for electronic devices.

Kirchhoff's Rules for Electric Circuit Analysis

Electric circuits generally consist of a number of resistors and cells interconnected, sometimes in a complicated way. The formulas derived earlier for series and parallel combinations of resistors are not always sufficient to determine all the currents and potential differences in the circuit.

Applying Kirchhoff's Rules

Two rules, called Kirchhoff’s rules, are very useful for the analysis of electric circuits. Given a circuit, we start by labeling currents in each resistor with a symbol, say I, and a directed arrow to indicate that a current I flows along the resistor in the indicated direction. If I is ultimately determined to be positive, the actual current in the resistor is in the direction of the arrow. If I turns out to be negative, the current actually flows in a direction opposite to the arrow.

Similarly, for each source (i.e., cell or some other source of electrical power), the positive and negative electrodes are labeled, as well as a directed arrow with a symbol for the current flowing through the cell. This will tell us the potential difference, V = V(P) – V(N) = ε – I r.

Application: The Wheatstone Bridge

As an application of Kirchhoff’s rules, consider the circuit known as the Wheatstone bridge. The bridge has four resistors: R₁, R₂, R₃, and R₄.

Components and Connections

• Across one pair of diagonally opposite points (A and C in the figure), a source is connected. This (i.e., AC) is called the battery arm.
• Between the other two vertices, B and D, a galvanometer G (which is a device to detect currents) is connected. This line, shown as BD in the figure, is called the galvanometer arm.

For simplicity, we assume that the cell has no internal resistance. In general, there will be currents flowing across all the resistors as well as a current I_g through G.

Balanced Bridge Condition

Of special interest is the case of a balanced bridge where the resistors are such that I_g = 0. We can easily get the balance condition, such that there is no current through G. In this case, Kirchhoff’s junction rule is applied to junctions D and B.