DC Circuit

Electromotive Force

Electromotive force is the work done by an energy source like cell, battery, generator, etc. to send unit charge in a circuit. Let’s dive into more detail explanation of electromotive force (emf) starting from potential difference.

Potential Difference

The work done in bringing the unit charge from one point to another point in the external circuit is called potential difference between the two points. \[\text{p.d.}=\frac{\text{Work Done}}{\text{Charge Transferred}}\] \[V=\frac{W}{q}\]

The potential difference between two points has only one value because p.d. is independent of path.

[Potential Difference Between Two Points]

Electromotive Force (emf)

Alessandro Volta invented the cell and he was the first person who introduced the term ‘electromotive force’. He defined electromotive force as a force required to separate positive and negative charges i.e. moved and hence, the term “electromotive”.

Later, Maxwell used emf for electromotive potential difference. It does not represent force; in fact, emf represents energy per unit charge. Hence, the word “force” in “electromotive force” is a misnomer.

All conductors offer resistance to the flow of charge, so work is needed to be done to maintain the current through the conductor. This work is done by the energy sources like cell, battery, generator etc. Such source is called source of emf or seat of emf.

Definition of Electromotive Force (emf)

The work done by the source to send unit charge in the whole circuit is called emf. \[E\text{ (emf)}=\frac{W}{q}\]

Its SI unit is joule per coulomb $(JC^{-1})$ or volt.

The emf of a source is said to be one volt if one joule of work is done by it to flow one coulomb of charge in the whole circuit.

In terms of potential difference, emf can be defined as the potential difference between the terminals of a source of current when no current is drawn from it (or, when the circuit is open).

Electromotive force of a source of current depends on the nature of the source. For example, the emf of a chemical cell depends on the nature of electrodes, nature and the concentration of electrolyte used in the cell and its temperature.

The emf of a source is divided into the potential differences across the resistors. This is because emf is the work done per unit charge for whole circuit while potential difference is the work done per unit charge across each resistor. If $V_1, V_2, V_3 …$ are the potential difference across the resistors in series then, the emf of the source is given by, \[E=V_1+V_2+V_3+…\]

[Also See: Parallel Combination of Resistors]

Terminal Potential Difference

The potential difference between the poles of the cell in closed circuit (when current is drawn from the cell) is called terminal potential difference of the cell.

Internal Resistance of a Cell

When the terminals of a cell are joined by a wire, the positive ions flow from the lower potential to the higher potential. The, other ions and neutral atoms of electrolyte of cell also offer resistance to their flow. Hence, the resistance offered by a cell is called internal resistance of the cell. It is denoted by r.

The internal resistance of a cell is considered to be connected in series with the cell. Due to the internal resistance, the emf of the cell is not equal to the terminal potential difference. An ideal cell is that cell which has zero internal resistance which is not possible in practice.

The value of internal resistance of the cell depends upon;

  1. Surface area of the cell. Larger is the surface area, less is the internal resistance.
  2. Distance between the electrodes. Larger is the distance, larger is the internal resistance.
  3. Nature, concentration and temperature of the electrolyte solution.
  4. Nature of the electrodes.

Circuit Formula

Consider a circuit as shown in figure.

Circuit formula to find the internal resistance of a cell (Electromotive Force)

Clearly, the total resistance of the cell is $R+r$. Then, from Ohm’s law, \[\text{Total Current}=\frac{\text{Total emf}}{\text{Total Resistance}}\] \[I=\frac{E}{R+r}\] \[E=IR+Ir \text{ __(1)}\]

If $V$ is the p.d. across the resistance $R$, then, \[V=IR \text{ __(2)}\] From $(1)$ and $(2)$, \[E=V+Ir\] \[E-V=Ir\]

Potential drop $(E-V)$ is the voltage that is lost or used by the internal resistance of the cell. \[r=\frac{E-V}{I}\] \[r=\frac{E-V}{\frac{V}{R}}\] \[r=\left(\frac{E-V}{V}\right)R\]

This relation is known as circuit formula which can be used to calculate the internal resistance of the cell.

When there is no current in the circuit i.e. the circuit is open, then, \[E=V\]

Hence, in an open circuit, terminal potential difference is equal to the emf of the cell.

During discharging a cell i.e. when the cell delivers charges in the circuit, \[E=V+Ir\] Therefore, in this case, the emf is greater than the potential difference.

During charging a cell, the direction of current reverses, therefore, \[E=V+(-Ir)\] \[V=E+Ir\] In this case, the terminal difference is greater than the potential difference.

The conventional direction of current is directed from positive terminal to negative terminal outside the cell (in the circuit) and from negative terminal to positive terminal inside the cell.