Question

Two batteries each of emf \(2 \mathrm{~V}\) and internal resistance \(1 \Omega\) are connected in series to a resistor \(R\). Maximum Possible power consumed by the resistor \(=\ldots .\) (A) \(3.2 \mathrm{~W}\) (B) \((16 / 9) \mathrm{W}\) (C) \((8 / 9) \mathrm{W}\) (D) \(2 \mathrm{~W}\)

Short answer

The maximum possible power consumed by the resistor is \(2 \mathrm{~W}\), which corresponds to option (D).

Step-by-step solution

Calculate the total EMF and internal resistance of the circuit

The batteries are connected in series, so the total EMF of the circuit is the sum of the emf of both batteries, and the total internal resistance is the sum of their internal resistances. Total EMF (E_total) = E_1 + E_2 = 2V + 2V = 4V Total internal resistance (r_total) = r_1 + r_2 = 1Ω + 1Ω = 2Ω

Apply the maximum power theorem

To maximize the power consumed by the resistor, we need to find the value of R when the power is at its maximum. According to the maximum power transfer theorem, this occurs when the load resistance (R) is equal to the internal resistance (r_total) of the circuit. So, R = r_total = 2Ω

Calculate the total current in the circuit

Now, let's find the total current in the circuit (I_total) using Ohm's law: I_total = E_total / (R + r_total) = 4V / (2Ω + 2Ω) = 4V / 4Ω = 1A

Calculate the maximum power consumed by the resistor

Finally, let's calculate the maximum power (P_max) consumed by the resistor using the formula P = I^2 * R: P_max = I_total^2 * R = (1A)^2 * 2Ω = 1A * 2Ω = 2W So the maximum possible power consumed by the resistor is 2 W, which corresponds to option (D).

Key concepts

Series Circuit

In a series circuit, all the components are connected end-to-end in a single path for the electric current. This means that the same current flows through all components sequentially.
Each component can include resistors, batteries, or other elements like capacitors or inductors.
In the context of the given problem, two batteries are connected in series which results in the cumulative addition of their electromotive forces (EMF) and internal resistances.
The calculated total EMF becomes the sum of each individual battery's EMF, and the total internal resistance is the sum of each battery's internal resistance.

• Current Flow: The current is the same through each component in the circuit because there is only one path for it to travel.
• Voltage Addition: The voltage provided to the circuit is the total EMF, so the voltages of series-connected batteries add up.
• Application in Problems: In series circuits like the problem above, understanding how to add voltages and resistances is crucial to harness the total potential energy available.

Series circuits are simple and function effectively, particularly when one wishes to add voltages or ensure the same current flows through several components.

EMF and Internal Resistance

Electromotive Force (EMF) and internal resistance are fundamental properties of a battery. The EMF of a battery refers to the energy provided per coulomb of charge as it passes through a source, often measured in volts (V).
However, a real battery always has some internal resistance which limits the efficiency of energy transfer.
This resistance causes some energy to be lost within the battery itself.

• EMF: Think of EMF as the ideal voltage a battery can provide without internal resistance loss. It reflects the battery's full potential.
• Internal Resistance: Present in real-world batteries, this parameter accounts for energy lost as heat, diminishing the voltage that effectively powers external loads.
• Series Circuit Interaction: In series circuits, internal resistances add arithmetically just like resistors would.
  So, understanding cumulative internal resistance is pivotal when dealing with multiple voltages in series.

In the example problem, the internal resistance adds directly to affect how the applied voltage behaves across the external circuit, impacting the calculation of maximum power transfer.

Ohm's Law

Ohm's Law is a cornerstone of electrical circuits, describing the relationship between voltage, current, and resistance.
This principle is expressed in the formula:
\[ V = I \times R \]where:

• \( V \) is the voltage across the component,
• \( I \) is the current flowing through it, and
• \( R \) is the resistance of the component.

In the process of calculating maximum power transfer, Ohm's Law helps determine the current flowing in the circuit once the resistance values are established.

• Determining Current: By applying this law, the problem calculates the current as 1 ampere, given the total equivalent resistance and total EMF.
• Influence on Power Calculation: Ohm's Law allows for the derivation of power formulas like \( P = I^2 \times R \), which was employed in determining the power consumed by the resistor in the given problem.
• Significance: Understanding how to shift between these fundamental electrical quantities is crucial for analyzing any circuit and solving related problems.
  It's a guiding principle to troubleshoot and forecast electric responses in varied conditions.

This concept not only facilitates theoretical calculations but also supports practical decisions in circuit design and debugging.