Problem 23 - Master Resource Book in JEE Main Physics

Question

A resistor of \(10 \mathrm{k} \Omega\) having tolerance \(10 \%\) is connected in series with another resistor of \(20 \mathrm{k} \Omega\) having tolerance 20\%. The tolerance of the combination will be approximately (a) 109 (b) \(13 \%\) (c) \(17 \%\) (d) \(20 \%\)

Step-by-Step Solution

Answer

The tolerance of the combination is approximately 17%.

Step 1: Understand the Problem

We are given two resistors in series with their resistances and tolerances. One resistor is 10kΩ with 10% tolerance, and the other is 20kΩ with 20% tolerance, and we need to determine the tolerance of the combination.

Step 2: Calculate the Combined Resistance

The total resistance in a series combination is simply the sum of the individual resistances. So, we add the resistances of the two resistors: \[R_1 = 10 ext{k}\,\Omega, \quad R_2 = 20 ext{k}\,\Omega\]\[R_{total} = R_1 + R_2 = 10 ext{k}\,\Omega + 20 ext{k}\,\Omega = 30 ext{k}\,\Omega\]

Step 3: Calculate Tolerance of Each Resistor

Calculate the tolerance range for each resistor:For the 10kΩ resistor: \[ T_1 = 10 ext{k}\,\Omega \times 0.10 = 1 ext{k}\,\Omega \]For the 20kΩ resistor:\[ T_2 = 20 ext{k}\,\Omega \times 0.20 = 4 ext{k}\,\Omega \]

Step 4: Determine the Total Tolerance Range

Since the resistors are in series, the maximum tolerance range is simply the sum of the individual tolerances:\[T_{total} = T_1 + T_2 = 1 ext{k}\,\Omega + 4\text{k}\,\Omega = 5 ext{k}\,\Omega\]

Step 5: Calculate the Tolerance of the Combined Resistor

The tolerance of the combination as a percentage is given by dividing the total tolerance by the total resistance:\[\%T = \frac{T_{total}}{R_{total}} \times 100\%= \frac{5\text{k}\,\Omega}{30\text{k}\,\Omega} \times 100\% = 16.67\%\]

Step 6: Choose the Closest Option

The calculated tolerance of the combination is approximately 16.67%. We compare this value with the given options: 109, 13, 17, and 20. The closest option to 16.67% is 17%.

Key Concepts

Series Circuit

A series circuit is one of the fundamental types of electrical circuits. In a series circuit, components, like resistors, are connected end-to-end in such a way that there is only one path for current to flow. This means the current that passes through one component will also flow through the subsequent components in the circuit. It is important to remember that in a series circuit:

• The total resistance is simply the sum of the resistances of each component.
• The same current flows through each component.
• The total voltage across the circuit is the sum of the voltages across each component.

This is particularly useful when calculating resistances, as we can just add up the resistances of individual resistors to find the total equivalent resistance. For example, for two resistors of 10kΩ and 20kΩ connected in series, the total resistance would be 30kΩ. This simple rule simplifies many calculations and is a key feature of series circuits.

Resistor Combination

Understanding how to combine resistors is essential in designing and analyzing circuits. Resistors can be combined in two primary configurations: series and parallel. Here, we focus on resistors in series.

When resistors are in series, the total resistance, as mentioned, is the sum of individual resistances. The formula for combining resistors in series is:\[R_{total} = R_1 + R_2 + R_3 + ... + R_n\]where \(R_{total}\) is the total resistance, and \(R_1, R_2, R_3, ..., R_n\) are the resistances of individual resistors. This series combination means each resistor adds its resistance to the overall circuit, making it easy to manage and predict how a circuit will behave under different loads. In our example, a 10kΩ and a 20kΩ resistor in series gives us a total resistance of 30kΩ. The simple addition makes analysis straightforward.

Tolerance Calculation

Calculating the tolerance of a combination of resistors is crucial for understanding the variability and reliability of an electrical circuit. Tolerance indicates how much the actual resistance can vary from the stated resistance value, expressed as a percentage.

For individual resistors:

• You multiply the resistance by the tolerance percentage to get the tolerance range.
• For a 10kΩ resistor with a 10% tolerance, the tolerance range would be 1kΩ.
• For a 20kΩ resistor with a 20% tolerance, the tolerance range would be 4kΩ.

To find the tolerance of resistors in series, you add up the individual tolerances to find the total tolerance range:\[ T_{total} = T_1 + T_2 = 1\text{k}\,\Omega + 4\text{k}\,\Omega = 5\text{k}\,\Omega \]Then, to express this as a percentage of the total resistance:\[\%T = \left( \frac{T_{total}}{R_{total}} \right) \times 100\%\]For our series resistors, the calculation becomes:\[\%T = \left( \frac{5\text{k}\,\Omega}{30\text{k}\,\Omega} \right) \times 100\% = 16.67\%\]This result shows the tolerance expresses how accurately the established resistance can be relied on in practice. In our given choices, 17% is the closest match to this calculated tolerance, indicating it as the best answer. Understanding tolerance helps optimize circuits for their intended use and reliability.