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## 'More Realistic Electric Kettle' printed from http://nrich.maths.org/

An electric kettle is a quite simple household appliance which is used everyday to convert electrical energy to heat energy. The diagram shows a circuit which provides a simple model for what happens in the electric kettle.

The kettle diagram: V is the voltage of the power supply (a power supply in UK has 240 volts (V)), R is the resistance of a heating element measured in ohms ($\Omega$), I is a current flowing around the circuit in
amps (A).

The experiment was carried out to investigate the relationship between the resistance of heating element and the temperature which settle down after a while. Results are shown in the table.

**Resistance R/$\Omega$** |
**Temperature T/°C** |

65 |
97 |

100 |
70 |

150 |
53 |

200 |
45 |

250 |
40 |

- Why does the temperature settle down after a while?
- Plot the graph of data for the temperature against the resistance.
- Could you recongise the function? Using points (100, 70) and (250, 40) find the equation of this function. (Hint: the function is of the form T = T
_{0} + C/R where T_{0} and C are constants.)
- Find the temperature of the room where the experiment was conducted.
- Does this equation is suitable for all values of resistance?

In practise we know that for an electric kettle to boil water takes 3-4 minutes. Could you

investigate how the time needed to boil water is related with the amount of water which are you heating?

Extension: the power of the kettle is given by the formula where R is the resistance of the heating element, r is the internal resistance and V is the voltage of power supply. $$P = \frac{RV^2}{(r+R)^2}$$
Suppose it is given that V = 240 volts, r = 5 ohms. Find R such that we get the maximum power.

Try to deduce generally the value of R for which the power is the biggest if the internal resistance is r.

*An old electric kettle picture is taken from http://www.sciencemuseum.org.uk/images/I059/10325939.aspx*