Analyzing Resistors in Parallel Combination

The question describes a circuit with two resistors, $R_1$ and $R_2$, connected in parallel. We are given specific information about their physical properties: they are made of the same material and have the same thickness. This means they have the same resistivity ($\rho$) and the same cross-sectional area ($A$). We are also told that the length of $R_2$ is twice the length of $R_1$, which can be written as $L_2 = 2L_1$. We need to find the relationship between the total resistance ($R$) of the parallel combination and the individual resistances.

Understanding Electrical Resistance

The resistance of a conductor is determined by its material, length, and cross-sectional area. The formula for resistance is:

$$R = \rho \frac{L}{A}$$

where:

• $R$ is the resistance.
• $\rho$ (rho) is the resistivity of the material.
• $L$ is the length of the conductor.
• $A$ is the cross-sectional area of the conductor.

Relating $R_1$ and $R_2$

Given that $R_1$ and $R_2$ are made of the same material and have the same thickness (same cross-sectional area $A$), their resistances can be written as:

• For $R_1$: $R_1 = \rho \frac{L_1}{A}$
• For $R_2$: $R_2 = \rho \frac{L_2}{A}$

We know that $L_2 = 2L_1$. Substituting this into the expression for $R_2$:

$$R_2 = \rho \frac{2L_1}{A}$$

We can rewrite this as:

$$R_2 = 2 \times \left( \rho \frac{L_1}{A} \right)$$

Since $R_1 = \rho \frac{L_1}{A}$, we can substitute $R_1$ into the equation for $R_2$:

$$R_2 = 2R_1$$

So, the resistance of $R_2$ is twice the resistance of $R_1$.

Calculating Total Resistance in Parallel

For two resistors connected in parallel, the total resistance $R$ is given by the formula:

$$\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}$$

Alternatively, the total resistance can be calculated using the product-sum formula:

$$R = \frac{R_1 R_2}{R_1 + R_2}$$

Finding the Relationship between Total Resistance and $R_1$

We found that $R_2 = 2R_1$. Now, substitute this relationship into the parallel resistance formula $R = \frac{R_1 R_2}{R_1 + R_2}$:

$$R = \frac{R_1 (2R_1)}{R_1 + 2R_1}$$

Simplify the expression:

$$R = \frac{2R_1^2}{3R_1}$$

Cancel out one $R_1$ term from the numerator and the denominator:

$$R = \frac{2R_1}{3}$$

To match the format of the options, we can rearrange this equation by multiplying both sides by 3:

$$3R = 2R_1$$

Comparing with Options

Let's compare our derived relationship $3R = 2R_1$ with the given options:

• Option 1: $3R = 2R_1$
• Option 2: $3R = 2R_2$
• Option 3: $2R = 3R_1$
• Option 4: $2R = 3R_2$

Our result $3R = 2R_1$ matches Option 1.

Revision Table: Resistance Concepts

Additional Information: Factors Affecting Resistance and Circuit Types

Understanding the factors affecting resistance and how components behave in series versus parallel is crucial for circuit analysis.

• Factors Affecting Resistance: The resistance of a conductor depends on four main factors:
  • Material: Different materials have different resistivities. Good conductors like copper have low resistivity, while insulators like rubber have high resistivity.
  • Length (L): Resistance is directly proportional to length. A longer wire offers more resistance.
  • Cross-sectional Area (A): Resistance is inversely proportional to the cross-sectional area. A thicker wire (larger area) offers less resistance.
  • Temperature: For most metallic conductors, resistance increases with increasing temperature.
• Series Circuits: Components are connected end-to-end, forming a single path for current.
  • Current is the same through all components.
  • Total voltage is the sum of individual voltage drops.
  • Total resistance is the sum of individual resistances ($R_s = R_1 + R_2 + \dots$).
• Parallel Circuits: Components are connected across each other, providing multiple paths for current.
  • Voltage is the same across all components.
  • Total current is the sum of currents through individual branches.
  • Total resistance is calculated using the reciprocal formula ($\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \dots$). The total resistance in parallel is always less than the smallest individual resistance.