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Chapter 18: Problem 55

\(n\) identical cells, each of emf \(E\) and internal resistance \(r\), are connected in series a cell \(A\) is joined with reverse polarity. The potential difference across each cell, except \(A\) is (a) \(\frac{2 n E}{n-2}\) (b) \(\frac{(n-2) E}{n}\) (c) \(\frac{(n-1) E}{n}\) (d) \(\frac{2 E}{n}\)

Short Answer

Expert verified
The potential difference across each cell, except \(A\), is \(\frac{2E}{n}\).

Step by step solution

01

Understand the Problem Setup

We have a total of \(n\) cells, each with emf \(E\) and internal resistance \(r\), connected in series. One cell, denoted as \(A\), is reversed in polarity. We need to find the potential difference across each cell except for \(A\).
02

Calculate the Total EMF

Since cell \(A\) is reversed, its emf will work against the emf of the other cells. The combined emf of the series, taking into account the reverse cell, is \((n-1)E - E = (n-2)E\).
03

Calculate the Total Internal Resistance

The total internal resistance of the series connection is simply the sum of the internal resistances: \(nr\).
04

Apply Ohm's Law to Calculate the Total Current

Using the formula \(V = IR\), where \(V\) is the potential, \(I\) is the current, and \(R\) is the resistance, the total current \(I\) in the circuit can be calculated using \((n-2)E = I \cdot nr\). This simplifies to \(I = \frac{(n-2)E}{nr}\).
05

Determine Potential Difference Across Each Cell Except A

For each cell (not including \(A\)), the potential difference \(V\) is given by \(IR_j - Ir\) (where \(IR_j\) is the potential due to emf and \(Ir\) is the potential drop due to resistance). This simplifies to \(V = E - Ir = E - \frac{(n-2)E}{nr}\cdot r = E - \frac{(n-2)E}{n}\).
06

Simplify to Find the Final Potential Difference

Further simplifying the expression from Step 5: \(V = E - \frac{(n-2)E}{n} = \frac{nE}{n} - \frac{(n-2)E}{n} = \frac{2E}{n}\).

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Key Concepts

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

Internal Resistance
Internal resistance is an essential concept in understanding how real-world batteries and cells operate.It refers to the opposition or resistance within the power source itself, affecting the performance and efficiency of the cells.
Unlike an ideal battery, which delivers its entire electromotive force (EMF) without any energy loss, real batteries have some internal resistance that causes a voltage drop when delivering current.
This internal resistance is typically denoted by the symbol \( r \), and in a practical scenario, it influences the cell's output voltage.
  • The higher the internal resistance, the greater the voltage drop within the battery, reducing the effective voltage available to the external circuit.
  • In series connections, like in our exercise, the total internal resistance is the sum of the internal resistances of each cell.
  • This total internal resistance plays a crucial role in determining the current flowing through the circuit as it opposes the external load.
Understanding internal resistance helps in estimating how much energy a battery can effectively supply to a circuit. It is a key factor when calculating the actual output voltage and current in circuits, especially when figuring out battery performance over time.
Series Circuit
A series circuit is one of the simplest ways to connect multiple electrical components, such as batteries or resistors.In this configuration, each component is connected end-to-end, forming a single pathway for the electric current to flow.
In the context of our exercise, we have \( n \) identical cells connected in series. This setup affects the way EMF and internal resistance are calculated for the circuit.
  • In a series circuit, the total EMF is the sum of the EMFs of each individual component.
  • Similarly, the total internal resistance is the sum of the internal resistances of all components in the series.
  • This means that the series circuit is simple to calculate in terms of total voltage and resistance, making it a straightforward model for understanding basic circuit principles.
However, reversing the polarity of one of these cells, as we see in the exercise, means subtracting its EMF from the total.This adjustment is crucial for solving the problem and calculating the correct potential difference across each cell.
Ohm's Law
Ohm's Law is a fundamental principle in electronics, establishing a relationship between voltage, current, and resistance.
The law is commonly expressed with the equation \( V = IR \), where \( V \) is the voltage across the circuit, \( I \) is the current flowing through the circuit, and \( R \) is the resistance.
Ohm's Law is vital in circuits with series connections because it helps to determine how the total voltage and resistance affect the current.
  • When dealing with a series circuit, the total current in the circuit can be calculated by applying Ohm's Law, taking into account the total EMF and total internal resistance.
  • In our exercise, the formula rearranges to find \( I \), or the current, using \( I = \frac{(n-2)E}{nr} \), after considering the reversed cell's effect on the total EMF.
  • By knowing the current, we can determine the potential drop across each component, as shown in the step-by-step solution.
Ohm's Law provides the simplicity needed for calculating complex scenarios by applying a systematic approach to voltage, resistance, and current interactions in electrical circuits.

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Most popular questions from this chapter

An electric heater of \(1.08 \mathrm{~kW}\) is immersed in water. After the water has reached a temperature of \(100^{\circ} \mathrm{C}\), how much time will be required to produce \(100 \mathrm{~g}\) of steam? (Latent heat of steam \(=540 \mathrm{calg}^{-1}\) )

A heating element using nichrome connected to a \(230 \mathrm{~V}\) supply draws an initial current of \(3.2 \mathrm{~A}\) which settles after a few seconds to a steady value of \(2.8 \mathrm{~A}\). What is the steady temperature of the heating element, if the room temperature is \(27.0^{\circ} \mathrm{C}\) ? Temperature coefficient of resistance of nichrome averaged over the temperature range involved is \(1.70 \times 10^{-4}{ }^{\circ} \mathrm{C}^{-1} .\) (a) \(967^{\circ} \mathrm{C}\) (b) \(867^{\circ} \mathrm{C}\) (c) \(853^{\circ} \mathrm{C}\) (d) \(937^{\circ} \mathrm{C}\)

The electron of hydrogen atom is considered to be revolving round a proton in circular orbit of radius \(\hbar^{2} / m e^{2}\) with velocity \(e^{2} / \hbar\) where \(\hbar=h / 2 \pi\). The current \(i\) is (a) \(\frac{4 \pi^{2} m e^{5}}{h^{2}}\) (b) \(\frac{4 \pi^{2} m e^{2}}{h^{3}}\) (c) \(\frac{4 \pi^{2} m^{2} \mathrm{e}^{2}}{h^{3}}\) (d) \(\frac{4 \pi^{2} m e^{5}}{h^{3}}\)

In a Wheatstone bridge, \(P=90 \Omega, Q=110 \Omega, R=40 \Omega\) and \(S=60 \Omega\) and a cell of emf \(4 \mathrm{~V}\). Then the potential difference between the diagonal along which a galvanometer is connected, is (a) \(-0.2 \mathrm{~V}\) (b) \(+0.2 \mathrm{~V}\) (c) \(-1 \mathrm{~V}\) (d) \(+1 \mathrm{~V}\)

The equivalent resistance of \(n\) resistors each of same resistance when connected in series is \(R\). If the same resistances are connected in parallel, the equivalent resistance will be (a) \(R / n^{2}\) (b) \(R / n\) (c) \(n^{2} R\) (d) \(n R\)
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