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Chapter 19: Problem 34

If \(t_{1}\) and \(t_{2}\) are the time taken by two different coils for producing same heat with same supply, then the time taken by them to produce the same heat when connected in parallel will be: (a) \(\frac{t_{1} t_{2}}{t_{1}+t_{2}}\) (b) \(t_{1}+t_{2}\) (c) \(t_{1} t_{2}\) (d) \(\frac{t_{1}+t_{2}}{t_{1} t_{2}}\)

Short Answer

Expert verified
The time taken for parallel connection is \(\frac{t_1 t_2}{t_1 + t_2}\). Option (a).

Step by step solution

01

Understand the Problem

We need to find the time taken for two coils connected in parallel to produce the same amount of heat as they did individually, when powered separately and taking the same time, \(t_1\) and \(t_2\).
02

Recall the Heat Generation Formula

The heat generated by a resistor (or coil in this case) is given by the formula \(Q = i^2 R t\), where \(Q\) is the heat, \(i\) is the current, \(R\) is the resistance, and \(t\) is the time. Since they produce the same heat individually, we deduce that \(i_1^2 R_1 t_1 = i_2^2 R_2 t_2 = \text{same heat}\).
03

Set the Same Heat Condition for Two Parallel Coils

When the coils are in parallel, the same voltage is applied across them. The condition for the same heat in each coil changes to \(i_1 R_1 t = i_2 R_2 t = \text{same heat}\) because \(i_1 + i_2\) is the current through them parallel.
04

Calculate the Equivalent Resistance for Parallel Connection

For parallel resistors, the equivalent resistance \(R_{eq} = \left(\frac{1}{R_1} + \frac{1}{R_2}\right)^{-1}\).
05

Apply the Heat Formula to the Parallel Connection

The total heat produced in the parallel configuration for a time \(t\) is \((i_1 + i_2)^2 R_{eq} t\). Set this equal to the single coil heat generation \(i_1^2 R_1 t_1\) or \(i_2^2 R_2 t_2\) and solve for \(t\).
06

Derive the Formula for Equivalent Time

To create an equation for the time by ensuring the same total heat, use derived symbol replacement for sums of resistances, solve the equal heat time equation, and coinsolve by simplification: \((i_1^2 R_1 t_1 + i_2^2 R_2 t_2)\), leading to \(\frac{t_1 t_2}{t_1 + t_2}\).
07

Select the Correct Answer

Based on our derived formula, \(t_{parallel} = \frac{t_1 t_2}{t_1 + t_2}\). This matches option (a).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parallel resistors
When resistors are connected in parallel, they share the same voltage across them. This is different from a series connection, where each resistor would have the same current passing through it. In a parallel circuit:
  • The total current flowing in the circuit is the sum of currents through each individual resistor.
  • Since the resistors are arranged differently, their total resistance, known as the equivalent resistance, is reduced.
  • This reduction in resistance allows for more total current to flow from the voltage source, thereby affecting the heat dissipation.
To calculate the equivalent resistance for resistors in parallel, the formula is:\[R_{eq} = \left(\frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n}\right)^{-1}\]Understanding this concept helps in analyzing how different resistors will behave when connected in such configurations, especially regarding heat generation and energy consumption.
Heat formula in physics
The heat generated in an electrical circuit, often referred to as Joule heating, can be determined using the formula:\[Q = i^2 R t\]where:
  • \(Q\) is the heat (in joules)
  • \(i\) is the current through the circuit (in amperes)
  • \(R\) is the resistance (in ohms)
  • \(t\) is the time for which the current flows (in seconds)
This formula indicates that the heat produced is proportional to the square of the current, the resistance, and the time the current flows through the resistor. In practical situations, this heat can be essential for things like electric heaters or detrimental as in overheating electronic components. Recognizing the relationship between these factors is critical for effectively managing power in any electronic device.
Time calculation in circuits
In the context of circuits, time calculation refers to evaluating how long it takes for certain electrical events to occur, such as reaching a certain amount of heat production or a specific current flow. When working with circuits that include resistors in parallel:
  • The calculation of time taken for certain processes changes due to shared voltage and varied current distribution.
  • As exemplified in our problem, the time, when the coils are in parallel, can be calculated by rearranging the heat formula and equivalent resistance equations.
For the specific exercise:
  • By equating the heat from both individual and parallel configurations, we use the formula:\[t_{parallel} = \frac{t_1 t_2}{t_1 + t_2}\]
  • This formula was derived based on the condition that both coils, when connected in parallel, generate the same amount of heat as when they are independent.
Understanding time calculation in such contexts is critical, ensuring correctness in electrical circuit applications and optimizing device performance.

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Most popular questions from this chapter

The thermo emf produced in a thermocouple is 3 microvolts per degree centigrade. if the temperature of the cold junction is \(20^{\circ} \mathrm{C}\) and the thermo \(\mathrm{emf}\) is \(0.3 \mathrm{mil}-\) livolts, the temperature of the hot junction is (a) \(80^{\circ} \mathrm{C}\) (b) \(100^{\circ} \mathrm{C}\) (c) \(120^{\circ} \mathrm{C}\) (d) \(140^{\circ} \mathrm{C}\)

Two ends of a conductor are at different temperatures. The electromotive force generated between the two ends is: (a) Thomson electromotive force (b) Peltier electromotive force (c) Seebeck electromotive force (d) none of the above

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Two electric bulbs have tungsten filaments of same length. If one of them gives 60 watts and the other 100 watts, then: (a) 100 watt bulb has thicker filament (b) 60 watt bulb has thicker filament (c) both filaments are of same thickness (d) it is impossible to get different wattage unless the lengths are different

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