Topic 4 – EMF and Internal Resistance

Batteries (or more strictly speaking cells) convert chemical energy into electrical energy.  Generators turn kinetic energy into electrical energy.  In doing so, they keep the negative terminal with an excess of electrons and the positive terminal with a deficiency of electrons.  A battery does a job of work in pumping the electrons around the circuit.  Positive charges do not move.

The early day physicists got it wrong when they said that electric current flows from positive to negative.  They didn't know about electrons.  When the mistake was discovered, they decided to stick to the positive to negative, so all conventional current flows from positive to negative.

A battery is said to produce Emf (electromotive force) which is defined as

the energy converted into electrical energy when unit charge passes through the source.

This is similar to the definition for potential difference that we saw before, except that it describes the conversion to electrical energy, rather than the conversion from electrical energy.  It represents the total energy that can be supplied to a circuit.  EMF is a voltage.

Note that EMF is energy per unit charge, NOT a force, which can lead to confusion.  Watch out for this particular bear trap.

A good working definition of emf is the open circuit terminal voltage of the battery, i.e. when there is no current flowing.  Although the old text books had a complex method for measuring emf using a metre bridge, nowadays a digital multimeter will give you a good reading as it takes a very small current indeed.

The energy supplied to a circuit by a battery is given by:

Where 

• W is the energy in J
• Q is the charge in C
• , curly E is the physics code for emf.

No circuit at all is 100 % efficient.  Some energy is dissipated in the wires, or even in the battery itself.

Internal Resistance

All batteries and generators dissipate heat internally when giving out a current, due to internal resistance.  A perfect battery has no internal resistance, but unfortunately there is no such thing as a perfect battery.  Nickel-Cadmium and Lead-Acid batteries have very low internal resistance, and we can regard these as almost perfect.  These batteries can provide very high currents.

Suppose we connect a cell to a high resistance voltmeter.  (A perfect voltmeter has infinite resistance.  A digital multimeter has a very high resistance, so needs a tiny current; it is almost perfect.  An ordinary moving coil voltmeter has a relatively low resistance, so it takes a small but appreciable current.)

In this circuit the voltmeter reads (very nearly) the emf.

Suppose we now add a load.  We will assume the wires have negligible resistance.

This time we find that the terminal voltage goes down to V.  Since V is less than , this tells us that not all of the voltage is being transferred to the outside circuit; some is lost due to the internal resistance which heats the battery up. 

  Emf = Useful volts  + Lost volts

  In code:

So we can represent the circuit as:

So our cell is now a perfect battery in series with an internal resistor, r.  You cannot open up the battery to find the internal resistor; it is part and parcel of the battery.

We can now treat this as a simple series circuit and we know that the current, I, will be the same throughout the circuit.  We also know the voltages in a series circuit add up to the battery voltage.

            Emf = voltage across R + voltage across the internal resistance

                =         V                        +                     v

We also know from Ohm’s Law that V = IR and v = Ir, so we can write:

= IR + Ir

Þ = I(R + r)

Many students panic at the sight of internal resistance problems.  All you have to do is turn the cell with the internal resistance into a perfect battery in series with its internal resistor, and treat it as a simple series circuit.