Combination of Cells in Series and in Parallel

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What are Cells?

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A cell is a device that uses chemical energy to generate electricity and maintains the flow of charge in a circuit. The energy is obtained from the chemical reactions that take place within these cells. Cells are commonly referred to as batteries, which are an essential day to day product required for a number of tasks. It is important to note that a battery is a collection of cells. And when placed in different circuits (in series or parallel), it functions differently. 
 

A cell consists of electrodes and electrolytes. The electrodes pass current in the circuit as they are conductors. The electrode which has higher potential is called anode or positive terminal of the cell. The electrode which has lower potential is called cathode or negative terminal of the cell.

A cell supplies energy when it is connected to an external load. The flow of current outside the cell is always from anode to cathode, while inside the cell the current flows from cathode to anode.

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Electromotive Force or EMF

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EMF or electromotive force is defined as the potential difference between the positive and negative electrodes when there is no current in the cell or when the cell is in an open circuit. The flow of the current in the cell is initiated by the EMF of the cell.

The SI unit of EMF is Volts (V). The formula of EMF is- 

ε = E/Q

Where, ε = emf (Volts), E= Energy (Joules) and Q= charge (Coulombs).

The formula of EMF can also be written as,

ε = IR + Ir

Where,

ε = EMF (Volts),

I = Current (Ampere),

R = Load resistance in ohms,

r is the internal resistance of a cell in ohms.

The video below explains this:

Electromotive Force Detailed Video Explanation:

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Internal Resistance

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Internal resistance is defined as the resistance offered by the electrolyte and electrodes when the current flows in the circuit. The flow of the current is opposed by the electrolytes and electrodes. The internal resistance is denoted by the letter ‘r’.

Let's consider a cell having two electrolytes which are connected to each other by an external resistance which is denoted by ‘R’. The current will flow from cathode to anode.

Closed Circuit: V = V₁ + V₂ – Ir 

Where (V₁+V₂) = initial potential difference,

Ir = potential drop across the internal resistance.

Therefore V = ε – Ir

Here ε=Emf, V=potential difference between 2 electrodes or it is the voltage drop across external resistance ‘R’.

⇒ IR = ε – Ir

Only when the circuit is closed the internal resistance will play its role. It varies from cell to cell.It is negligible where ε >> Ir.

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EMF and Internal Resistance Equation

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The emf denoted as ε and the internal resistor with resistance r which is connected in series modifies the cell. The resistance R with an external load resistor is also connected across the circuit. The potential difference, V across the cell is represented as:

V = V₊ + V – Ir

This represents the voltage drop present due to the internal resistance.

We know that 

ε = V₊ + V_-  = I(R + r)

ε = IR + Ir

= V + Ir

V = ε – Ir

Hence, V = ε – Ir

Here V is the potential difference across the circuit, ε is the emf, I is the current flowing through the circuit and r is internal resistance. 

In most instances, the internal resistance of a cell is not considered because ε >> Ir. Internal resistance usually changes in its value from cell to cell

Read More: Resistors in Series, Resistors in Parallel

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Cells in Series and Parallel

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There are mainly two types of Connections of cells i.e. series and parallel. Cells can be connected in series, parallel or a combination of both. 

In a series circuit, electrons can travel only in one path. Here, the current passing through each resistor will be the same. However the voltage across resistors in a series connection will be different in each resistor. Thus circuits connected in series ensure that they do not overheat easily. 

In a parallel circuit electrons travel in several branches. In the case of a parallel circuit, the voltage remains the same across each resistor in the circuit and the cells can be arranged in different forms. Hence parallel circuits are used as a current divider and are easy to connect or disconnect a new cell or other component. This does not affect the other elements in the circuit. However, it uses a lot of wires and that is why the circuit becomes complex.

Cells in Series

When multiple cells are arranged in a manner where the positive terminal from the first cell is connected to the negative terminal of the next cell, and this sequence continues, it is called a series. Specifically, this arrangement is known as a series combination.

The arrangement is as shown below:

Let the EMF of the first cell = ε₁ and EMF of the second cell = ε₂.

Consider 3 points A, B and C as shown in the figure. The potential at point A= V_A, potential at point B=V_B and potential at point C=V_C. 

Let us assume that the circuit is closed. As a result, there will be some flow of current. Resistance will be offered by the cell itself. 

Internal resistance of first cell =r₁ and Internal resistance of second cell=r₂.

Equation for potential difference ε₁(first cell):-

For an open circuit

Potential difference between points A and B (V_A – V_B) = potential difference between the 2 electrodes.

Therefore VA – VB =  ε₁ – Ir₁… Eq (i)

Where Ir₁ = voltage drop across the internal resistance.

Similarly for equation for potential difference E₂(second cell):-

VB – Vc =  ε₂ – Ir₂ ...Eq (ii)

Adding (i) and (ii), 

VA – Vc = ε₁ – Ir₁ + ε₂ – Ir₂

VA – Vc = ε₁ + ε₂ – I(r₁ + r₂) ....Eq (iii)

Assuming if there are only 2 cells then the EMF = ε_equivalent= (ε₁ + ε₂) and resistance=r_equivalent= (r₁+r₂).

Using (iii)

V_A - V_c = ε_equivalent–Ir_equivalent … Eq(iv)

By comparing RHS of equations (iii) and (iv), we get

ε_equivalent= (ε₁ + ε₂) and r_equivalent= (r₁+r₂)

For cells arranged in series

The sum of the individual EMFs is the equivalent EMF. The sum of the individual resistances gives the equivalent resistance.

Let n be the number of cells that are connected in series and connected across an external resistance R.

Internal resistance of each cell =r, Emf =ε

Total resistance = R +r_equivalent

Where R=external resistance of the external circuit.

Total resistance = R + nr where nr(sum of all the n resistances).

Thus the effective Emf of cells = nε, where nε (EMF of each cell).

Therefore current flowing through resistance

R(I) = \(\frac{Effective EMF}{Total Resistance} = \frac{nE}{R + nr}\)

Case 1: When R>>nr

Then I = \(\frac{nE}{R}\)  ignoring nr.

⇒ I = \(\frac{nE}{R}\)

Here (ε/R) = current caused by a single cell across external resistance.

So, the current becomes n times the current caused by the single cell.

Case 2: When 

R< I

I = \(\frac{n\varepsilon}{nR}\)

I = \(\frac{\varepsilon}{r}\)

Current present in the circuit is equal to the current due to a single cell across.

Conclusion

When cells are arranged in the series, the current flowing through the circuit depends upon the value of external and internal resistance. 

Cells in Parallel

A series of cells is said to be in parallel when all the positive terminals are connected together and all the negative terminals are connected together. 

In lieu of this arrangement, the potential difference across each of these cells will be the same.

Potential difference (first cell) =ε₁; potential difference (second cell) =ε₂.

Internal resistance (first cell) =r₁; Internal resistance (second cell) =r₂. 

The cell internal resistance is offered to the cell as current is flowing through it.

Equation for potential difference ε₁(first cell):-

Potential difference between 2 points A and B.

Therefore V = ε₁ – I₁r₁  (for first cell)

(Current I gets divided into 2 parts I₁ and I₂; I₁ across ε₁).

I₁ = (ε₁ – V)/r₁ …..Eq (i)

Equation for potential difference ε₂(second cell):

V = (ε₂ – I₂)r₂ (current I₂ flows across ε₂)

I₂ = (ε₂ – V)/r₂ …...Eq(ii)

Also, I = I₁ + I₂

Therefore using (i) and (ii)

I = (ε₂ – I₂)r₂ + (ε₂ – V)/r₂

⇒ ((ε₁/r₁) + (ε₂/r₂) – (V(1/r₁) – (1/r₂))

⇒ V = (I/(1/r₁) + (1/r₂)) + ((ε₁/r₁) + (ε₂/r₂) + (1/r₂))

V = I(r₁r₂/r₁ + r₂) + (ε₁r₂ + ε₂r₁)/r₁ + r₂)…….Eq(iii)

Consider only 1 cell instead of 2 cells, i.e. 

ε_equivalent= (ε₁ and ε₂) and resistance = r_equivalent = (r₁ and r₂).

Therefore V= ε_equivalent - Ir_equivalent ...Eq (iv)

From Equation (iii),

V = I / ((1/r₁) + (ε₂/r₁) + (ε₂r₂)) /  ((1/r₁) + (1/r₁))…….Eq (v)

After simplifying and Comparing (v) and (iv),

r_equivalent = (r₁ + r₂) / (r₁ + r₂), ε_equivalent = (ε₁r₁ + ε₂r₂)/r₁ + r₂)

Special Case

Let n be the number of cells connected in a parallel combination across an external resistance R. Let ε be the internal Emf of each cell and r be the internal resistance

Using ε_equivalent = \(\frac{(\varepsilon_1r_1 + \varepsilon_2r_2)}{r_1 + r_2}\) 

Putting r1 = r2 = r, ε₁ = ε₂ = ε; 

ε_equivalent = \(\frac{\varepsilon r}{n}\)

Total resistance=R + r_equivalent = \(\frac{r}{n}\) 

Current through R, I = (Effective Emf)/(total resistance) = (ε) / (R + (r/n)) = (nε) / (nR + r)

• Case 1: R<<r; I = nε/r
• Case 2: R>>>r; I = ε/r

Important Note:

• If cells are arranged in series, it increases the voltage.
• If cells are arranged in parallel, it increases the current.

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Kirchhoff's Laws Detailed Video Explanation:

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