Understanding Combined Resistance in Circuits

When electrical components like resistors are connected, their combined effect on the total resistance of the circuit depends on how they are arranged. There are two fundamental ways to connect resistors: in series and in parallel.

Series Connection of Resistances

In a series connection, resistors are connected end-to-end, forming a single path for the current to flow through. Imagine a chain where each link is a resistor; the current must pass through every resistor in sequence.

Key characteristics of resistors in series:

• The same current flows through each resistor.
• The total voltage across the series combination is the sum of the voltages across each resistor.
• The total or equivalent resistance (\(R_{eq}\)) is the sum of the individual resistances.

If resistors \(R_1, R_2, R_3, \dots, R_n\) are connected in series, the equivalent resistance is given by:

\(R_{eq} = R_1 + R_2 + R_3 + \dots + R_n\)

In a series connection, the total resistance is always greater than the largest individual resistance.

Parallel Connection of Resistances

In a parallel connection, resistors are connected across the same two points in a circuit. This arrangement provides multiple paths for the current to flow.

Key characteristics of resistors in parallel:

• The voltage across each resistor is the same.
• The total current flowing into the parallel combination is divided among the individual paths and is the sum of the currents through each resistor.
• The reciprocal of the total or equivalent resistance (\(R_{eq}\)) is the sum of the reciprocals of the individual resistances.

If resistors \(R_1, R_2, R_3, \dots, R_n\) are connected in parallel, the reciprocal of the equivalent resistance is given by:

\(\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots + \frac{1}{R_n}\)

A particularly useful formula for two resistors (\(R_1\) and \(R_2\)) in parallel is:

\(R_{eq} = \frac{R_1 \times R_2}{R_1 + R_2}\)

In a parallel connection, the total resistance is always less than the smallest individual resistance. This is because providing more paths for the current to flow effectively reduces the overall opposition to the current.

Analyzing the Question and Options

The question asks when a number of resistances are connected in _______, their combined resistance is less than the smallest individual resistance.

Let's look at the options:

• horizontal: This is not a standard electrical connection term for resistors.
• box: This is not a standard electrical connection term for resistors.
• series: As discussed, series connection results in a total resistance that is greater than the largest individual resistance. So, this option is incorrect.
• parallel: As discussed, parallel connection results in a total resistance that is less than the smallest individual resistance. This matches the condition described in the question.

Therefore, when resistances are connected in parallel, their combined resistance is less than the smallest individual resistance.

Revision Table: Series vs. Parallel Resistance

Additional Information: Applications of Series and Parallel Circuits

Understanding how resistances combine in series and parallel is crucial for designing and analyzing electrical circuits. Different configurations are used for different purposes:

• Series circuits: Often used in voltage dividers, Christmas tree lights (older type where if one bulb fails, all fail), or to increase the total resistance in a circuit.
• Parallel circuits: Common in household wiring (so that turning off one appliance doesn't affect others), current dividers, or to decrease the total resistance to handle more current or provide alternative paths. For example, multiple power outlets in a room are wired in parallel.

Choosing between series and parallel connections depends entirely on the desired outcome for the circuit's total resistance and current/voltage distribution.