Question

How would you arrange 48 cells each of e.m.f \(2 \mathrm{~V}\) and internal resistance \(1.5 \Omega\) so as to pass maximum current through the external resistance of \(2 \Omega\) ? (A) 2 cells in 24 groups (B) 4 cells in 12 groups (C) 8 cells in 6 groups (D) 3 cells in 16 groups

Short answer

The best arrangement to pass the maximum current through the external resistance of 2 Ω is 2 cells in 24 groups, which allows a maximum current of 45.18 A.

Step-by-step solution

Understanding the Arrangements

We have four possible arrangements of cells: (A) 2 cells in 24 groups (B) 4 cells in 12 groups (C) 8 cells in 6 groups (D) 3 cells in 16 groups In each arrangement, we will calculate the equivalent e.m.f (E) and internal resistance (r) for each group of cells.

Calculating Equivalent E and r for Each Arrangement

For cells connected in series, the equivalent E is the sum of the e.m.f of each cell, whereas the equivalent r is the sum of the internal resistances of all cells in the series. For cells connected in parallel, the sum of the reciprocals of E and r remains the same. (A) 2 cells in 24 groups: Here, 2 cells are connected in series and each group is connected in parallel. E = 2 + 2 = 4 V (E of cells connected in series) r = 1.5 + 1.5 = 3 Ω (r for cells connected in series) Equivalent E for the 24 groups connected in parallel = \(24 \times 4 = 96 \mathrm{~V}\) Equivalent r for the 24 groups connected in parallel = \(3/24 = 0.125 \mathrm{~ \Omega}\) (B) 4 cells in 12 groups: Here, 4 cells are connected in series and each group is connected in parallel. E = 2 + 2 + 2 + 2 = 8 V (E of cells connected in series) r = 1.5 + 1.5 + 1.5 + 1.5 = 6 Ω (r for cells connected in series) Equivalent E for the 12 groups connected in parallel = \(12 \times 8 = 96 \mathrm{~V}\) Equivalent r for the 12 groups connected in parallel = \(6/12 = 0.5 \mathrm{~ \Omega}\) (C) 8 cells in 6 groups: Here, 8 cells are connected in series and each group is connected in parallel. E = 8 * 2 = 16 V (E of cells connected in series) r = 8 * 1.5 = 12 Ω (r for cells connected in serie)s Equivalent E for the 6 groups groups connected in parallel = \(6 \times 16 = 96 \mathrm{~V}\) Equivalent r for the 6 groups connected in parallel = \(12/6 = 2 \mathrm{~ \Omega}\) (D) 3 cells in 16 groups: Here, 3 cells are connected in series and each group is connected in parallel. E = 2 + 2 + 2 = 6 V (E of cells connected in series) r = 1.5 + 1.5 + 1.5 = 4.5 Ω (r for cells connected in series) Equivalent E for the 16 groups connected in parallel = \(16 \times 6 = 96 \mathrm{~V}\) Equivalent r for the 16 groups connected in parallel = \(4.5/16 = 0.28125 \mathrm{~ \Omega}\)

Calculating the Maximum Current for Each Arrangement

We have the external resistance R = 2 Ω. We will use Ohm's law (I = E / (R + r)) to calculate the current for each arrangement. (A) I = \(96 / (2 + 0.125) = 96 / 2.125 = 45.18 \mathrm{~A}\) (B) I = \(96 / (2 + 0.5) = 96 / 2.5 = 38.4 \mathrm{~A}\) (C) I = \(96 / (2 + 2) = 96 / 4 = 24 \mathrm{~A}\) (D) I = \(96 / (2 + 0.28125) = 96 / 2.28125 = 42.09 \mathrm{~A}\)

Identifying the Best Arrangement

Comparing the currents for each arrangement, we see that the maximum current is achieved in arrangement (A) with 45.18 A. Therefore, the best arrangement to pass the maximum current through the external resistor of 2 Ω is 2 cells in 24 groups.

Key concepts

Electrical Circuit Arrangements

Electrical circuits involve various ways of connecting electric components such as cells, resistors, etc. Arranging these components strategically can change the circuit's behavior and performance. When connecting multiple cells to achieve maximum efficiency and desired voltage, there are typically options such as series and parallel arrangements. The problem of arranging cells often requires careful consideration of both the total electromotive force (e.m.f) and the total internal resistance that may affect the current flow through the circuit. The purpose of arranging cells, as seen in the exercise, is to optimize these factors to achieve a specific electrical outcome, like maximizing current through an external resistance.

Internal Resistance of Cells

Each cell in an electrical circuit has what's known as internal resistance, denoted usually by the symbol 'r.' Internal resistance can limit the flow of current by using up some of the energy produced by the cell itself. This matters most when connecting many cells together, such as in a complex circuit arrangement, as seen in the exercise. In theory, cells with lower internal resistance are preferable for achieving higher current output. When cells are connected in series, their internal resistances add up. Thus, the more cells you have in a series, the higher the total internal resistance. However, when they're connected in parallel, the total internal resistance decreases, which can be useful for minimizing losses and achieving a higher current through an external load.

Ohm's Law

Ohm's Law is a fundamental theorem in electrical engineering, aiding in understanding how voltage, current, and resistance are interrelated. The law states that the current (I) passing through a conductor between two points is directly proportional to the voltage (V) across the two points and inversely proportional to the resistance (R) of the conductor, expressed by the formula: \[ I = \frac{V}{R} \] In circuits involving cells with internal resistance, Ohm's Law is expanded to account for both internal (r) and external resistance (R). Thus, the formula can be restated as: \[ I = \frac{E}{R + r} \] Where 'E' is the total e.m.f. of the cells. This extended formula helps to calculate the maximum current that can pass through an external resistor, factoring in the internal resistance of the cells.

Series and Parallel Circuits

The arrangement of electrical components in series or parallel can significantly affect the circuit's overall performance. In a series circuit, components are arranged sequentially, so that the current passing through each component is the same, but the total voltage is the sum of individual voltages However, in a parallel circuit, all components share the same voltage across them, and the total current is the sum of the currents through each component. These arrangements can be mixed within more complex circuits to optimize parameters like e.m.f and resistance as seen in the exercise where multiple series and parallel groups of cells are connected. Choosing the right setup depends largely on what you hope to achieve, whether it's greater voltage or greater current, and involves understanding the interplay between collective e.m.f and total resistance.