Two resistors R 1 and R 2 arranged in parallel combination in an electrical closed circuit are made of the same material and of the same thickness. If the length of R 2 is twice the length of R 1, then the total resistance R satisfies

Question

Prepp Team · Accepted Answer

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

Concept	Description	Formula
Resistance (R)	Opposition to electric current flow.	$R = \rho \frac{L}{A}$
Resistivity ($\rho$)	Intrinsic property of a material indicating its resistance.	Unit: Ohm-meter ($\Omega \cdot m$)
Resistance in Series	Total resistance is the sum of individual resistances.	$R_{total} = R_1 + R_2 + \dots + R_n$
Resistance in Parallel	Reciprocal of total resistance is the sum of reciprocals of individual resistances.	$\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots + \frac{1}{R_n}$

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.