Electromotive Force (emf)

Let's start with a couple of real life examples related to electromotive force (emf). When you continuously throw a tennis ball upward, the ball does not continue its motion (rising and falling down) if you stop to throw the ball. Your hand that pushes the ball to through again acts like a pump to lift the ball from lower to higher potential energy.

Similarly in a water fountain, the water is being pumped from lower to higher potential energy, so that the fountain can continue its cycle. If there is not such a pump, there won't be any fountain. Similar happens in an electric circuit. If there is not a pump that lifts charges "uphill" from lower to higher potential energy to create a constant potential difference, there won't be any current in the circuit.

Figure 1 In a fountain the water is again pumped to higher potential energy and the fountain can continue its cycle.

If an electric field is applied in a conductor of a fixed length which is not a part of an electric circuit. The free electrons move until there is an opposite electric field canceling the applied electric field and there is no current in the conductor. The current is not possible in an incomplete circuit. So, for the steady current, there must be a complete electric circuit. If we have a complete circuit, there must be an electric field in the circuit otherwise there is no current.

As the charge flows in an electric circuit, its potential energy is lost. The lost potential energy is transferred to the atoms of current carrying material and spent in heating the material. In Figure 2 shown below, considering the wires have zero resistance, the potential energy decreases as the charges flow through the resistor of resistance R.

We often neglect the resistance of real wires, but real wires have resistance. Usually the resistance of real wires is negligible to the actual load resistance in an electric circuit and we neglect the resistance of wires in such cases.

The potential energy of a charge at a point in a circuit is the same as the potential energy when the charge reaches the same point again. So there must be a device which lifts charges from lower to higher potential energy. The device that acts as a pump to lift charges from lower to higher potential energy is called the source of emf. The most familiar example of the source of emf is a battery.

Figure 2 A source of emf (battery) is connected to a resistance in an electric circuit.

The source of emf has terminals of lower and higher potential energy. The positive terminal represented by $+$ sign is higher potential terminal and that represented by $-$ sign is lower potential terminal as shown in Figure 2 above. The electric field is provided by a fixed potential difference between the terminals of the source of emf.

So there must be a device (source of emf) in a circuit which provides the fixed potential difference in an electric circuit. In short the potential energy decreases as the charge flows in a circuit and there must be a part in the circuit which again lifts charges "uphill" so the charges can continue the cycle.

Electromotive force (emf) is not a force but it is the energy. The term electromotive force is inaccurate as it is not a force but it is the energy; it's like calling energy a force. Only for some historical reasons, we still have to say that name like in specific gravity which has nothing to do with gravity.

Electromotive force (efm) is the energy or potential difference or voltage which lifts charges from lower to higher potential energy.

A source which produces electromotive force (emf) is called the source of emf such as a battery. A battery acts like a charge pump in an electric circuit similar to a water pump in a fountain. Batteries convert chemical energy into electrical energy. The batteries do their work because of the oxidation-reduction reactions also called redox reactions. When electrons are transferred from an atom, that atom is said to be oxidized and the atom which gained electrons is said to be reduced.

A battery is one of the sources of emf not emf. The emf is the energy per unit charge (potential difference or voltage) to lift a charge against electric field. Electromotive force separates positive and negative charges and maintains a fixed potential difference across the terminals of a source of emf. Electromotive force allows us to convert other forms of energy such as mechanical, chemical, thermal etc. into electrical energy.

In an ideal source of emf, the potential difference across its terminals does not change in all conditions. In Figure 3 an ideal source of emf is shown (schematic diagram) which has two terminals, one is the terminal of higher potential (positive terminal) $a$ and another is the terminal of lower potential (negative terminal) $b$ as shown in Figure 3. When the charge moves from lower to higher potential within the source, that is from $b$ to $a$, the charge experiences two forces. One is the electrostatic force $\vec F_{\text{e}}$ which pulls the charge towards lower potential and another is the nonelectrostatic force $\vec F_{\text{ne}}$ which pushes the charge towards higher potential.

Figure 3 The work done by the upward force $\vec F_{\text{ne}}$ (nonelectrostatic force) per unit charge is the emf.

The nonelectrostatic force can result from anything and its origin depends on the source of emf. In a battery, the nonelectrostatic force results from chemical reactions. The work done by the nonelectrostatic force from lower to higher potential terminal within the source is the emf. In an ideal source of emf both forces balance each other; the charges move from lower to higher potential with constant velocity.

So anything that lifts charges uphill from lower to higher potential energy to create a fixed potential difference is emf. Electromotive force does not allow charges to flow across terminals within the source. The electrostatic force tries to draw charges from higher to lower potential but the non-electrostatic force does not allow the charges to fall towards the lower potential.

The direction of current within the source is from lower to higher potential opposite to the direction of current in a conductor. It's because we are lifting charges from lower to higher potential. Note that the direction of current is always considered as the direction of the flow of positive charge and current is not a vector which we know in electric current.

When a battery (source of emf) is connected to an electric circuit, the charge flows from position of higher potential to the position of lower potential and the potential energy continuously decreases. The decreased potential energy is transferred to the resistance of the circuit which increases the vibration of atoms and thus spent in heating the current carrying material . Note that the charge does not gain kinetic energy as it flows.

For an ideal source of emf the potential difference across the terminals does not change even if there is current in an electric circuit but that is not the case for real sources of emf. If there is current in an electric circuit, the potential difference (voltage) across the terminals of the source of emf decreases due to internal resistance within the source of emf, that is the charge also flows within the material of the source of emf and experiences internal resistance, so the real potential difference is always less than the emf.

Electromotive force (emf) is the potential difference across the terminals of its source when there is no current in an electric circuit.

If you have a battery of $1.5 \text{V}$ which is not connected to any complete circuit, the value $1.5 \text{V}$ is the emf. When that battery is connected to a complete circuit, the actual voltage is not $1.5 \text{V}$ but it is always less than that value due to the internal resistance within the battery. The potential difference across the terminals of a source of emf when there is current in an electric circuit is called terminal voltage. And that terminal voltage is always less than the emf (the voltage when there is no current in an electric circuit).

Figure 4 An electric circuit with a source of emf and load resistance.

In an electric circuit shown in Figure 4 above if $I$ is the current and $r$ is the internal resistance, the potential drop within the source of emf is $Ir$. If $R$ is the external resistance in the circuit, the potential difference across the resistance, that is potential at $a$ with respect to the potential at $b$ is $V_{\text{ab}} = IR$, so the emf $\mathcal{E}$ is

\[\mathcal{E} = V_{\text{ab}} + Ir = IR + Ir = I(R + r)\]

The above equation shows us that the current depends on the total resistance of a circuit, that is $I = \mathcal{E}/(R+r)$. Greater the resistance, lesser the current. The terminal voltage $V_{\text{ab}}$ is the voltage across the external resistance $R$ also called load resistance and it is equal to the voltage across the terminals of the source (across the terminals of source means including the internal resistance, otherwise in above figure you may exclude the internal resistance) when there is current in the circuit. The resistance of connecting wires is considered to be negligible, so the load resistance can be provided by either a resistor or an electric device.

Figure 4 A graph of how potential changes around a circuit.

A graph of how potential changes when when a source of emf is connected to form a complete electric circuit is shown above. Let's move along the circuit starting at point $a$ clockwise in the loop. The charge ends its cycle at the point $a$ (at the positive terminal). The source of emf does work on the charge and maintains a fixed potential difference between terminals $a$ and $b$. It means the potential increases from $a$ to $b$, and the potential of $a$ with respect to the potential of $b$, that is the potential difference is equal to the emf $\mathcal {E}$.

The potential decreases across the internal resistance $r$. If $I$ is the current in the circuit, the potential difference across internal resistance (voltage drop) within the source of emf is $Ir$, that is the potential decreases by an amount $Ir$ in the internal resistance. Considering the wires have zero resistance, there is no voltage drop from $d$ to $e$. The potential decreases by the amount $IR$ as we move across resistance $R$ from $e$ to $f$. The resistance $R$ is our actual load resistance of our circuit.

Some examples of the sources of emf are batteries (chemical to electrical), electric generators (mechanical to electrical), photo-voltaic cells(sunlight to electrical) etc. The thermal energy released by burning fuels can also be converted to electrical energy. Thermal energy converts water to steam which then runs turbines to run electrical generator to generate electricity.

Electromagnetism
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