Equivalence of Heat Dissipation in External Resistors: An Insight into EMF, Resistance, and Internal Resistance

Brief Introduction to the Problem

Consider an EMF source (electric generator or battery) with electromotive force (EMF) e and internal resistance r. This source is connected first to an external resistance R1 and then to an external resistance R2 for the same period of time. The question is to find out the value of the internal resistance r if the total heat dissipated in both R1 and R2 is the same.

Solving the Problem Step by Step

Step 1: Understanding the Problem

The heat dissipated in any resistor can be calculated using Joule's law: H I^2 R t, where H is the heat, I is the current, R is the resistance, and t is the time. We need to ensure that the total heat dissipation in R1 and R2 is the same.

Step 2: Considering the Circuit with Internal Resistance

When an EMF source with internal resistance r is connected to an external resistor R, the current in the circuit can be determined by I frac{e}{R r}. The power dissipated in the external resistor R can be calculated as P_R I^2 R left(frac{e}{R r}right)^2 R.

Step 3: Heat Dissipated in R1

When the EMF source is connected to R1, the current and the heat dissipated can be calculated as:

I_1 frac{e}{R_1 r}

H_1 I_1^2 R_1 left(frac{e}{R_1 r}right)^2 R_1

Step 4: Heat Dissipated in R2

Similarly, when the EMF source is connected to R2, the current and the heat dissipated can be calculated as:

I_2 frac{e}{R_2 r}

H_2 I_2^2 R_2 left(frac{e}{R_2 r}right)^2 R_2

Step 5: Equating the Heat Dissipation

As the heat dissipated in both resistors is the same, we can set H1 equal to H2:

left(frac{e}{R_1 r}right)^2 R_1 left(frac{e}{R_2 r}right)^2 R_2

frac{R_1}{(R_1 r)^2} frac{R_2}{(R_2 r)^2}

frac{R_1}{(R_1 r)^2} frac{R_2}{(R_2 r)^2}

(R_1)(R_2 r)^2 (R_2)(R_1 r)^2

R_1 R_2 2 R_1 r r^2 R_2 R_1 2 R_2 r r^2

2 R_1 r 2 R_2 r

2 R_1 2 R_2

r frac{R_1 R_2}{2}

Conclusion and Final Answer

Therefore, the internal resistance r of the EMF source must satisfy the condition r frac{R_1 R_2}{2} to ensure that the heat dissipated in both R1 and R2 is the same.

Key Terms and Concepts

EMF (Electromotive Force)

EMF is a measure of the electrochemical potential difference between two points of a circuit when no current is flowing. It is the energy provided by an external source to move a charge through a conductor. The unit of EMF is the volt (V).

Internal Resistance

Internal resistance is the resistance offered by the internal parts of a power source such as a battery. It dissipates energy and reduces the effective voltage applied to the external circuit. Internal resistance can significantly affect the performance and efficiency of the power source.

Heat Dissipation in Resistors

Heat dissipation in resistors is a consequence of the electrical power being converted into thermal energy. It is calculated using Joule's law: H I^2 R t. Understanding heat dissipation is crucial for designing circuits to ensure they do not overheat.

Practical Applications and Further Reading

The principles discussed in this article can be applied to a variety of circuits and devices, from simple electrical appliances to complex electronic systems. By understanding the relationship between EMF, internal resistance, and heat dissipation, engineers can optimize the performance of circuits and power sources.

Further Reading

All About Circuits: Electromotive Force and Voltage
Offers a detailed explanation of EMF and voltage drop in circuits. Electronicstutorials World: Heat Dissipation in Electrical and Electronic Circuits
Provides insights into the mechanisms of heat dissipation in different types of circuits. Electrical4U: Internal Resistance of a Cell or Battery
Details the significance and calculation of internal resistance in cells and batteries.