When electrical components like resistors are connected in a circuit, the way they are connected affects the total or equivalent resistance of the circuit. There are two primary ways to connect resistors: in series and in parallel.

Understanding Parallel Resistors

In a parallel connection, resistors are connected across the same two points in a circuit. This means that the voltage across each resistor is the same, but the current may divide among them. The total resistance in a parallel circuit is calculated differently than in a series circuit.

The formula for calculating the total equivalent resistance ($R_{total}$) when resistors are connected in parallel is given by the reciprocal of the sum of the reciprocals of individual resistances. For resistors $R_1, R_2, R_3, \dots, R_n$ connected in parallel, the formula is:

\begin{equation} \frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots + \frac{1}{R_n} \end{equation}

Calculating Total Resistance for Three Equal Resistors in Parallel

The question describes a specific scenario: three equal resistors connected in a parallel configuration. Let's denote the resistance of each individual resistor as $R$. Since the three resistors are equal, we have $R_1 = R_2 = R_3 = R$.

Now, we can apply the parallel resistance formula for three resistors:

\begin{equation} \frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \end{equation}

Substitute the individual resistance $R$ into the formula:

\begin{equation} \frac{1}{R_{total}} = \frac{1}{R} + \frac{1}{R} + \frac{1}{R} \end{equation}

To sum the fractions on the right side, since they have a common denominator ($R$), we add the numerators:

\begin{equation} \frac{1}{R_{total}} = \frac{1+1+1}{R} \end{equation}

\begin{equation} \frac{1}{R_{total}} = \frac{3}{R} \end{equation}

To find the total resistance ($R_{total}$), we take the reciprocal of both sides of the equation:

\begin{equation} R_{total} = \frac{R}{3} \end{equation}

This result shows that when three equal resistors with resistance $R$ are connected in parallel, the total equivalent resistance is $R$ divided by 3. In other words, the total resistance is one-third of the individual resistance.

Summary of Resistance Calculation

Let's summarize the calculation:

• Individual resistance of each of the three equal resistors = $R$
• Connection type = Parallel
• Formula for parallel resistance: $\frac{1}{R_{total}} = \sum \frac{1}{R_i}$
• Applying the formula: $\frac{1}{R_{total}} = \frac{1}{R} + \frac{1}{R} + \frac{1}{R} = \frac{3}{R}$
• Solving for $R_{total}$: $R_{total} = \frac{R}{3}$

Therefore, the total resistance in the circuit becomes one-third of the individual resistance.

Comparing Parallel vs. Series Resistance

It is helpful to compare this result with resistors connected in series. When resistors $R_1, R_2, R_3, \dots$ are connected in series, the total resistance ($R_{total}$) is simply the sum of the individual resistances:

\begin{equation} R_{total} = R_1 + R_2 + R_3 + \dots \end{equation}

If three equal resistors with resistance $R$ were connected in series, the total resistance would be $R_{total} = R + R + R = 3R$. This is three times the individual resistance.

This comparison highlights a key difference: connecting resistors in parallel decreases the total resistance, while connecting them in series increases the total resistance.

Revision Table - Electrical Resistance Calculations

Additional Information - Factors Affecting Resistance

While this question focuses on how connection type affects total resistance, the resistance of an individual resistor itself depends on several factors:

• Material: Different materials have different resistivity ($\rho$). Conductors like copper have low resistivity, while insulators have high resistivity.
• Length: The resistance ($R$) of a conductor is directly proportional to its length ($L$). Longer conductors have higher resistance. $R \propto L$.
• Cross-sectional Area: The resistance ($R$) of a conductor is inversely proportional to its cross-sectional area ($A$). Thicker conductors have lower resistance. $R \propto \frac{1}{A}$.
• Temperature: For most conductors, resistance increases with increasing temperature.

These factors are related by the formula for the resistance of a uniform conductor:

\begin{equation} R = \rho \frac{L}{A} \end{equation}

Understanding these factors helps in designing circuits and selecting appropriate components.