Electro Motive Force and Circuit Formula
ElectroMotive Force
When two terminals of a battery are connected by a conductor, an electric current flows through a conductor. One terminal continuously sends electrons into the conductor, while the other continuously receives electrons from it. In this way, the battery behaves like an electric pump. The property of a cell that makes the charge move in a particular direction is called electromotive force or emf.
The emf of a cell is the energy supplied by the cell to move unit charge round a circuit join to it. It is the potential difference between the terminals of the cell in an open circuit. It does not depend on the size of a cell but depends on the chemical used in it. It is denoted by E.
Internal Resistance
The resistance offered by a source for a current to pass through it is called its internal resistance. It is denoted by r. The internal resistance of a cell is the resistance offered by the electrolyte between the electrode of the cell when an electric current passes through it.
Internal resistance of a cell depends on many factors:
 The distance between the electrodes: r increases with increase in distance between the electrodes.
 The nature of the electrolyte: greater the conductivity of the electrolyte lesser is the internal resistance.
 Nature of electrodes.
 The area of the plates or electrodes immersed in the electrolyte; internal resistance decreases with increase in the area of the plates immersed.
 The temperature of the electrolyte: internal resistance decreases with rise of temperature.
Terminal Potential Difference
The terminal potential difference of a cell is defined as the potential difference between the positive and negative electrodes of a cell when the cell sends a current in an external circuit. It is measured in volt and is denoted by V. The terminal potential difference depends on the external resistance connected to a closed circuit.
Circuit Formula
It is a relation between emf, terminal potential difference and internal resistance of a cell. Consider a cell of emf E and internal resistance r connected to a resistance R in a circuit as shown in the figure. A steady current I flows through the circuits. Since p.d. across R is IR and potential difference across internal resistance r is Ir, then
\begin{align*} E &= IR + Ir \\ \text {or,} \: E &= I(R + r) \\ \text {or,} \: I &= \frac {E}{R + r} \dots (i) \\ \end{align*} This is the current in the circuit.As the external resistance R is connected across the terminals of the cell, so, terminal potential difference, V = Potential difference across R \begin{align*}\text {or,} \: V &= IR \dots (ii) \\ \text {Again,} \\ E &= IR + Ir = V + Ir \\ \text {or,} \: V &= E – Ir \dots (iii) \\ \end{align*}
This is a useful formula for terminal potential difference.
In an open circuit
\begin{align*}\\\: I = 0 \: \text {so,} \\ V &= E – 0r = E \\ \end{align*}
So, the emf of the cell in an open circuit is equal to its terminal p.d.
The current in the circuit can be obtained as
\begin{align*}\\ I &= \frac {E – V}{r} \dots (iv) \\ \text {From equations} \: (iv) \: \text {and}\: (iv), \: \text {we have} \\ \frac {E}{R + r} &= \frac {E – V}{r} \\ \text {or,} \: \frac {E}{E – V} &= \frac {R + r}{r} = \frac Rr + 1 \\ \text {or,} \: \frac {E}{E – V} – 1 &= \frac Rr \\ \text {or,} \: \frac {V}{E – V} &= \frac Rr \\ \text {or,} \: r &= \frac {E – V}{V} R \\ \end{align*}
This is the relation between emf, terminal p.d. and internal resistance of a cell. Knowing E, V and R, the value of r can be calculated.
Emf and terminal p.d. of a cell during charging and discharging
When a cell of emf E and internal resistance r is connected to a circuit, a current I flows in the circuit. This is called the discharging of the cell. In this condition terminal potential difference V is given by
\begin{align*} E &= V + Ir \\ \text {or,} \: V &= E – Ir \\ \end{align*}
The terminal p.d. of a cell during the cell by an external source as shown in the figure. In such condition, the current flows through the circuit against the emf of the cell and voltage is dropped in the internal resistance. So, terminal p.d V is given as
$$ V = E + Ir $$
That is, the terminal p.d. of the cell is greater than its emf during the charging of the cell.
Reference
Manu Kumar Khatry, Manoj Kumar Thapa,et al.Principle of Physics. Kathmandu: Ayam publication PVT LTD, 2010.
S.K. Gautam, J.M. Pradhan. A text Book of Physics. Kathmandu: Surya Publication, 2003.
Emf is the potential difference between the terminals of the cell in an open circuit.
The resistance offered by a source for a current to pass through it is called its internal resistance.
The terminal potential difference of a cell is defined as the potential difference between the positive and negative electrodes of a cell when the cell sends a current in an external circuit.
Circuit formula is a relation between emf, terminal potential difference and internal resistance of a cell.

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