Physics A Level
Chapter 8: Electric current 8.6 Electrical power
Physics A Level
Chapter 8: Electric current 8.6 Electrical power
The rate at which energy is transferred is known as power. Power P is measured in watts (W). (If you are not sure about this, refer back to Chapter 5, where we looked at the concept of power in relation to forces and work done.)
$power = \frac{{energy\,transferred}}{{time\,taken}} \equiv P = \frac{{\Delta W}}{{\Delta t}}$
where P is the power and ΔW is the energy transferred in a time ${\Delta t}$.
Take care not to confuse W for energy transferred or work done with W for watts.
Refer back to the equation derived from the definition of potential difference:
$V = \frac{{\Delta W}}{{\Delta Q}}$
This can be rearranged as:
$\Delta W = V\,\Delta Q$
Thus:
$P = \frac{W}{{\Delta t}} = \frac{{V\,\Delta Q}}{{\Delta t}} = V\left( {\frac{{\Delta Q}}{{\Delta t}}} \right)$
The ratio of charge to time, ${\frac{{\Delta Q}}{{\Delta t}}}$, is the current I in the component. Therefore:
$P = VI$
By substituting from the resistance equation $V = IR$, we get the alternative equations for power:
$P = {I^2}R$ and $P = \frac{{{V^2}}}{R}$
Questions
17) Calculate the current in a $60 W$ light bulb when it is connected to a $230 V$ power supply.
18) A power station supplies electrical energy to the grid at a voltage of $25 kV$. Calculate the output power of the station when the current it supplies is $40 kA$.
Power and resistance
A current I in a resistor of resistance R transfers energy to it. The resistor dissipates energy heating the resistor and the surroundings.. The p.d. V across the resistor is given by $V = IR$. Combining this with the equation for power, $P = VI$, gives us two further forms of the equation for power dissipated in the resistor:
$P = {I^2}R$
$P = \frac{{{V^2}}}{R}$
Which form of the equation we use in any particular situation depends on the information we have available to us. This is illustrated in Worked examples $7a$ and $7b$, which relate to a power station and to the grid cables that lead from it (Figure 8.15).
heat in these cables? (See Worked examples $7a$ and $7b$.)
Questions
19) A calculator is powered by a $3.0 V$ battery. The calculator’s resistance is $20\,k\Omega $. Calculate the power transferred to the calculator.
20) An energy-efficient light bulb is labelled ‘$230 V$, $15 W$’. This means that when connected to the $230 V$ mains supply it is fully lit and changes electrical energy to heat and light at the rate of $15 W$.
Calculate:
a: the current in the bulb when fully lit
b: its resistance when fully lit.
21) Calculate the resistance of a $100 W$ light bulb that draws a current of $0.43 A$ from a power supply
Calculating energy
We can use the relationship for power as energy transferred per unit time and the equation for electrical power to find the energy transferred in a circuit.
Since:
$power = current \times voltage$
and:
$energy = power \times time$
we have:
$\begin{array}{l}
energy\,transferred = current \times voltage \times time\\
W = IV\,\Delta t
\end{array}$
Working in SI units, this gives energy transferred in joules.
Questions
22) A $12 V$ car battery can supply a current of $10 A$ for 5.0 hours. Calculate how many joules of energy the battery transfers in this time.
23) A lamp is operated for $20 s$. The current in the lamp is $10 A$. In this time, it transfers $400 J$ of energy to the lamp. Calculate:
a: how much charge flows through the lamp
b: how much energy each coulomb of charge transfers to the lamp
c: the p.d. across the lamp.
EXAM-STYLE QUESTIONS
1) A small immersion heater is connected to a power supply of e.m.f. of $12 V$ for a time of $150 s$. The output power of the heater is $100 W$.
What charge passes through the heater? [1]
A: $1.4 C$
B: $8.0 C$
C: $1250 C$
D: $1800 C$
2) Which statement defines e.m.f.? [1]
A: The e.m.f. of a source is the energy transferred when charge is driven through a resistor.
B: The e.m.f. of a source is the energy transferred when charge is driven round a complete circuit.
C: The e.m.f. of a source is the energy transferred when unit charge is driven round a complete circuit.
D: The e.m.f. of a source is the energy transferred when unit charge is driven through a resistor.
3) Calculate the charge that passes through a lamp when there is a current of $150mA$ for 40 minutes. [3]
4) A generator produces a current of $40 A$. Calculate how long will it take for a total of $2000 C$ to flow through the output. [2]
5) In a lightning strike there is an average current of $30 kA$, which lasts for $2000\mu s$. Calculate the charge that is transferred in this process. [3]
6) a: A lamp of resistance $15\,\Omega $ is connected to a battery of e.m.f. $4.5 V$.
Calculate the current in the lamp. [2]
b: Calculate the resistance of the filament of an electric heater that takes a current of $6.5 A$ when it is connected across a mains supply of $230 V$. [2]
c: Calculate the voltage that is required to drive a current of $2.4 A$ through a wire of resistance $3.5\,\Omega $. [2]
[Total: 6]
7) A battery of e.m.f. $6 V$ produces a steady current of $2.4 A$ for 10 minutes.
Calculate:
a: the charge that it supplied [2]
b: the energy that it transferred. [2]
[Total: 4]
8) Calculate the energy gained by an electron when it is accelerated through a potential difference of $50 kV$. (Charge on the electron $ = - 1.6 \times {10^{ - 19}}C$.) [2]
9) A woman has available $1 A, 3 A, 5 A, 10 A$ and $13 A$ fuses. Explain which fuse she should use for a $120 V, 450 W$ hairdryer. [3]
10) This diagram shows the electrolysis of copper chloride.
a: i- On a copy of the diagram, mark the direction of the conventional current in the electrolyte. Label it conventional current. [1]
ii- Mark the direction of the electron flow in the connecting wires. Label this electron flow. [1]
b: In a time period of 8 minutes, $3.6 \times {10^{16}}$ chloride ($C{l^ - }$) ions are neutralised and liberated at the anode and $1.8 \times {10^{16}}$ copper ($C{u^{2 + }}$) ions are neutralised and deposited on the cathode.
i- Calculate the total charge passing through the electrolyte in this time. [2]
ii- Calculate the current in the circuit. [2]
[Total: 6]
11) This diagram shows an electron tube. Electrons moving from the cathode to the anode constitute a current. The current in the ammeter is $4.5 mA$.
a: Calculate the charge passing through the ammeter in 3 minutes. [3]
b: Calculate the number of electrons that hit the anode in 3 minutes. [3]
c: The potential difference between the cathode and the anode is $75 V$.
Calculate the energy gained by an electron as it travels from the cathode to the anode. [2]
[Total: 8]
12) A length of copper track on a printed circuit board has a cross-sectional area of $5.0 \times {10^{ - 8}}\,{m^2}$. The current in the track is $3.5 mA$. You are provided with some useful information about copper:
$1{m^3}$ of copper has a mass of $8.9 \times {10^3}\,kg$
$54 kg$ of copper contains $6.0 \times {10^{26}}\,atoms$
In copper, there is roughly one electron liberated from each copper atom.
a: Show that the electron number density n for copper is about ${10^{29}}\,{m^{ - 3}}$. [2]
b: Calculate the mean drift velocity of the electrons. [3]
[Total: 5]
13) a: Explain the difference between potential difference and e.m.f. [2]
b: A battery has negligible internal resistance, an e.m.f. of $12.0 V$ and a capacity of $100 A h$ (ampere-hours). Calculate:
i- the total charge that it can supply [2]
ii- the total energy that it can transfer. [2]
c: The battery is connected to a $27 W$ lamp. Calculate the resistance of the lamp. [2]
[Total: 8]
14) Some electricity-generating companies use a unit called the kilowatt-hour ($kWh$) to calculate energy bills. $1 kWh$ is the energy a kilowatt appliance transfers in 1 hour.
a: Show that 1 kWh is equal to $3.6 MJ$. [2]
b: An electric shower heater is rated at $230 V$, $9.5 kW$.
i- Calculate the current it will take from the mains supply. [2]
ii- Suggest why the shower requires a separate circuit from other appliances. [1]
iii- Suggest a suitable current rating for the fuse in this circuit. [1]
c: Calculate the energy transferred when a boy uses the shower for 5 minutes. [2]
[Total: 8]
15) A student is measuring the resistance per unit length of a resistance wire. He takes the following measurements.
| Quantity | Value | Uncertainty |
| length of wire | $80 mm$ | $ \pm 2\% $ |
| current in the wire | $2.4 A$ | $ \pm 1.0A$ |
| potential difference across the wire | $8.9 V$ | $ \pm 5\% $ |
a: Calculate the percentage uncertainty in the measurement of the current. [1]
b: Calculate the value of the resistance per unit length of the wire. [1]
c: Calculate the absolute uncertainty of the resistance per unit length of the wire. [2]
[Total: 4]
SELF-EVALUATION CHECKLIST
After studying the chapter, complete a table like this:
| I can | See topic… | Needs more work | Almost there | Ready to move on |
| understand of the nature of electric current | 8.2 | |||
| understand the term charge and recognise its unit, the coulomb | 8.2 | |||
| understand that charge is quantised | 8.2 | |||
| solve problems using the equation $ΔQ = IΔt$ | 8.2 | |||
| solve problems using the formula $I = nAve$ | 8.3 | |||
| solve problems involving the mean drift velocity of charge carriers | 8.3 | |||
| understand the terms potential difference, e.m.f. and the volt | 8.4 | |||
| use energy considerations to distinguish between p.d. and e.m.f. | 8.4 | |||
| define resistance and recognise its unit, the ohm | 8.5 | |||
| solve problems using the formula $V = IR$ | 8.5 | |||
| solve problems concerning energy and power in electric circuits. | 8.6 |