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# Internal resistance

## Batteries sweat if you make them work too hard

A battery converts chemical energy into electrical energy.  This conversion is caused by chemical reactions inside the battery.  The quicker the battery has to provide energy, the quicker those chemical reactions have to happen.

If you make the battery work hard then some of the chemical energy is converted into electrical energy and some into heat energy.  The heat energy makes the battery hot.

The harder you make the battery work, the more chemical energy is converted into heat energy and the hotter the battery gets.

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## How you can make a battery work hard

If you make a battery run a single component that demands energy very quickly, like a very bright bulb, or lots of components in parallel then the battery has to work very hard because it has to supply energy very quickly.

Remember batteries are (or try to be) constant voltage providers.  The current depends on the job they are doing.  When they work hard they provide a big current.  But as we'll see, if they provide a very big current then the voltage will drop.

## Less electrical energy means lower voltage

You can think of voltage as energy per charge.

If there is less electrical energy available (because some of the chemical energy ends up as heat in the battery) then the voltage across the terminals of the battery will drop.  This means the voltage available to the circuit also drops.

The practical upshot of this is that if you make a battery supply a big current by

• making it run something that demands a high current like a very bright (low resistance) bulb
• making it run lots of things at the same time (connected in parallel)
• shorting out the battery (by connecting one terminal to the other with a wire)

the voltage will drop.

## If electrical energy is converted into heat energy it must be a resistor

Batteries don't have this resistor inside them that you can take out and look at.  But they do have sources of resistance, for example the products of the chemical reactions and the metal parts with all their connections.

So batteries are often modelled as a perfect power supply (whose voltage never drops) in series with an imaginary resistor.

Obviously batteries are used to run something, like a bulb.  The resistance of the circuit they run is called the load resistance, RL.

The internal resistance is normally given the symbol r.  This doesn't mean that it's always small.  The resistance of a miniature watch battery might be 100 ohms or so.  A torch battery has an internal resistance of around 0.1 ohms and a car battery about 0.001 ohms.

Our imaginary internal resistor obeys Ohm's law just like any other resistor.  The only difference is that it's hidden inside the battery.

## Why is internal resistance important?

A 12 V car battery has exactly the same voltage as eight 1.5 V AA batteries.  Could you use these batteries to start you car?

The answer is a resounding no.  There are two ways of looking at it.

1. The AA batteries can't provide energy very quickly whereas your starter motor needs energy very quickly.
2. The internal resistance of your AA batteries is too high so the voltage drops from 12 V to close to 0 V as soon as you try and start your engine.

If you know you need to provide energy very quickly you need a very low internal resistance.  Just like a thick piece of wire, a big battery has a lower resistance.  So high power batteries need to be big, like a car battery.

If you don't need energy very quickly, in other words your device only draws a tiny current like a digital watch, then internal resistance is less important so you can afford to make your battery small.  This is useful if you want to fit it inside a watch!

## What is 'negligible internal resistance'?

This expression often appears in exam questions.

It doesn't mean the resistance has to be small per se.  It simply means that the battery isn't being made to work hard enough for its voltage to drop much.

## Electromotive force is the battery voltage when it's not running anything

The maximum voltage you can get from a battery is called the electromotive force or e.m.f.  It's called that for historical reasons but there's nothing special about it.  It's just a voltage.  It's normally given the symbol ε.

If internal resistance is not negligible and the battery's running something, like a bulb, then the voltage you actually measure across the terminals of the battery (and also across the bulb) will be lower than the e.m.f.  We call this lower voltage the load voltage, VL.

## Finding the internal resistance and e.m.f. of a battery

You can measure the e.m.f. of a battery by simply measuring the voltage across the terminals when it's not connected to anything.  This is called measuring the voltage in 'open circuit'.

You can't just measure the internal resistance directly because you can't get inside the battery.  So you have to do an experiment where you change the current drawn from the battery (by changing the load resistance) and measuring the p.d. across the terminals.

Kirchoff's voltage law says that if you add up the voltages across all the components in a series circuit it must exactly equal the battery voltage.

e.m.f. = voltage across internal resistance + voltage across load (e.g. a bulb)

In symbols this equation is

ε = Vinternal + VL

We know that V = IR, or using the appropriate terms for the internal resistance Vinternal = Ir, so

ε = Ir + VL

We can rearrange this equation to give

VL = -rI + ε

If you set up a circuit with a variable resistor for the load then you can change the current, I, drawn from the battery and measure the voltage across the terminals, VL.

Plotting VL against I gives a straight line with the size of the gradient equal to the internal resistance, r.  The e.m.f. is the intercept on the voltage axis, in other words, the voltage when the current is zero.

back to Lesson 7: Resistance and Ohm's Law