Electromotive Force (EMF)

Electromotive force is defined as the energy provided by a power source, like a battery or generator, to make electric charge flow through a circuit. Despite its name, the electromotive force is the energy per unit charge or potential difference created by the source.

1.0Basics of Electromagnetic Force EMF

Definition of EMF

• The term electromotive force, or EMF, describes the ability of a source (like a battery or generator) to push charges through a circuit.
• It is that energy delivered per unit charge, which is measured in joules per coulomb (J/C), hence equaling volts (V).
• Electromotive force refers to the voltage associated with the chemical reactions that take place within the battery, as it forces the charge through the circuit.

EMF can mathematically be represented as: $\epsilon =\frac{W}{Q}$

Where:

• ε is the EMF in volts,
• 𝑊 is the work done in joules,
• 𝑄 is the charge in coulombs.

The electromotive force unit is the volt, with one volt being the amount of energy needed to move one coulomb of charge with one joule of energy. This relationship is used in understanding how much energy would be required for the flow of current.

Factors Affecting EMF

• Chemical Reactions: The type and concentration of chemicals in the battery's electrolyte and the electrodes' material decide the EMF produced.
• Temperature: Temperature will affect the chemical reaction rates of the batteries; hence, temperature will also impact the EMF. High temperatures generally decrease internal resistance but accelerate the degradation process of the battery.
• Concentration of Electrolyte: The concentration of the electrolyte in electrochemical cells influences the voltage produced in them. Increasing the concentration causes an increase in EMF.
• Material Used for Electrodes: The composition of the material used for constructing electrodes in any battery or galvanic cell affects the value of EMF. The compositions vary in having different electrode potentials.
• Surface Area of Electrode: A bigger surface area for the electrodes while in contact with the electrolyte may influence EMF in any electrochemical cell.

The Formula For EMF In A Circuit

In a circuit with external resistance R and internal resistance r, the EMF can be calculated using the formula:

$\epsilon =I(R+r)$

Where:

ε - EMF in volts,

I - current in amperes,

R - external resistance in the circuit,

r - internal resistance of the power source.

Internal Resistance refers to the opposition to the flow of current within a source of EMF (electromotive force), such as a battery or a generator.

2.0Induced Electromotive Force

• Induced EMF: A change in magnetic flux through a coil or a conductor results in the production of an EMF, as discovered by Faraday's Law of Induction.

• The induced EMF, according to Faraday's Law, is proportional to the rate of change of magnetic flux through the circuit:

$\epsilon =-\frac{d{\varphi }_B}{dt}$

Where, 

• ${\varphi }_B$ is the magnetic flux
• $\frac{d{\varphi }_B}{dt}$ is the rate of change of magnetic flux

Induced Electromotive Force in Circuits

• An induced electromotive force results when a changing magnetic field is applied to a conductor or coil. This is the principle behind an electric generator and transformer.
• Faraday's Law of Induction states that a changing magnetic flux induces an EMF. This is also referred to as electromagnetic induction.

Combination of Batteries In The Circuit

Batteries can be combined in the circuit in two ways that is: 

1. Series Combination: 

In a series combination, batteries are connected end-to-end, i.e., the positive terminal of one battery is connected to the negative terminal of the next battery. The total EMF of the circuit will be the sum of all the EMFs of the individual batteries (E₁, E₂, E₃ …): 

E_Total = E₁ + E₂ + … + E_n.

And the total internal resistance will be: 

r_Total = r₁ + r₂ + … + r_n

2. Parallel Combination: 

In a parallel combination, all the positive terminals of the batteries are connected together as well and all the negative terminals are also connected together. The total EMF remains the same as that of  individual batteries (E₁, E₂, E₃ …), as the EMF is not added in parallel:

E_Total = E₁ = E₂ = E₃ … = E_n

The total internal resistance will be reduced to: 

$\frac{1}{r_{total}}=\frac{1}{r_1}+\frac{1}{r_2}+...+\frac{1}{r_n}$

3.0Motional Electromotive Force

Motional EMF is the EMF produced when a conductor is moving through a magnetic field. The cutting of the magnetic field lines by motion induces an EMF, according to Faraday's Law. The expression for motional EMF is:

$\epsilon =Blv$

Here, 

• B represents the magnetic field strength.
• l represents the length of the conductor,
• v represents the velocity of the conductor, which is perpendicular to the magnetic field. 

4.0Electromotive Force of a Battery 

• In a battery, chemical reactions in the battery that push electrons through an outer circuit make up the electromotive force. The EMF of a battery is steady when there is no drawn current.
• When the current flows, the internal resistance of the battery will decrease the potential difference between the terminals. The EMF is unchanged.

EMF and Internal Resistance

Every actual source of EMF, for example, a battery, has an internal resistance r, which opposes the flow of current. The relationship between the EMF $\epsilon$, current I, internal resistance r, and terminal voltage $V_t$ is given by:

      $V_t=\epsilon -I.r$

Terminal Potential Difference (V_t): Terminal Potential Difference measures the voltage seen across the two terminals of any source, battery, or generator with current being forced through its terminals. Hence, V_t is different from the electromotive force due to internal source resistance.

When the current flows, the internal resistance causes a voltage drop inside the source. So, the measured voltage at the terminals is less than the EMF.

5.0Solved Examples

Q1. A battery supplies 15 Joules of energy to drive 5 Coulombs of charge around a circuit. Calculate the electromotive force (EMF) of the battery.

Solution: We know that the formula for EMF is: $\epsilon =\frac{W}{Q}$

Where:

W= work done= 15 joules,

Q= charge= 5 coulombs.

Substitute the given values into the formula: $\epsilon$=155=3 Volts

Thus, the EMF of the battery is 3 Volts.

Q2.  Two batteries are connected in series. One battery has an EMF of 6 V and the other has an EMF of 9 V. What is the total EMF of the combination?

Solution: In a series connection, the EMF of the combination is the sum of the individual EMF values. Therefore:

${\epsilon }_{total}={\epsilon }_1+{\epsilon }_2$

Where:

$\epsilon =6Vand{\epsilon }_2=9V$

Substitute the values:

${\epsilon }_{total}=6+9+15V$

Thus, the total EMF of the combination is 15 Volts.

Q3: Two cells with EMFs E1=15 V and E2=10 V are connected in parallel. The internal resistances of the cells are r1=1 Ω and r2=2 Ω, respectively. Find the current supplied by each cell and the total current in the circuit if they are connected to an external load resistance of R=5 Ω.

Solution: We are given with EMF of cell 1, E1 = 15V, with internal resistance r1 = 1

EMF of Cell 2, E2 = 10V, Internal resistance r2 = 2 Ω

The current supplied by each cell can be calculated with the help of the following formula in a parallel connection: 

$I_1=\frac{E1-V}{r1},I_2=\frac{E2-V}{r2}$

The total current =

$I_{total}=\frac{V}{R}=I_1+I_2$

Using the Kirchhoff’s loop rule: 

$\frac{E1-V}{r1}+\frac{E2-V}{r2}=\frac{V}{R}$

$\frac{15-V}{1}+\frac{10-V}{2}=\frac{V}{5}$

$\frac{30-2V+10-V}{2}=\frac{V}{5}$

$(40-3V)5=2V$

$200–15V=2V$

$17V=200$

$V\approx 11.76V$

$I_1=\frac{15-11.76}{1}=3.24A,I_2=\frac{15-11.76}{2}=-0.88A$

$I_{total}=3.24-0.88=2.36A$