Electrical Power

What Is Electrical Power?

Electrical power is the rate at which electrical energy is transferred or transformed.

In other words, power tells us how quickly energy is being transferred.

The basic equation is:

P = E/t

where:

• P = power (W)
• E = energy transferred (J)
• t = time (s)

The SI unit of power is the watt (W).

1 W = 1 J/s

Therefore, a 100 W device transfers 100 J of energy every second.

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Power Is Not Energy

Power and energy are closely related, but they are not the same quantity.

Energy measures the total amount transferred.

Power measures how quickly that transfer occurs.

For example:

A 2000 W kettle transfers energy much faster than a 10 W LED lamp.

However, if the LED operates for a very long time, it can eventually transfer more total energy than the kettle.

The relationship is:

E = Pt

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Understanding the Watt

Because:

1 W = 1 J/s

a device rated at 500 W transfers:

500 J every second

A 2000 W heater transfers:

2000 J every second

or:

2 kJ every second

Higher power means a faster rate of energy transfer.

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Basic Power Calculation

A device transfers:

E = 6000 J

in:

t = 30 s

Power:

P = E/t

P = 6000/30

P = 200 W

The device transfers energy at a rate of 200 J each second.

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Electrical Power from Voltage and Current

Electrical energy is:

E = VIt

Since:

P = E/t

substitute:

P = VIt/t

Therefore:

P = VI

where:

• P = electrical power (W)
• V = potential difference (V)
• I = current (A)

This is one of the most important equations in electrical circuits.

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Example Using P = VI

An appliance operates at:

V = 230 V

I = 4 A

Power:

P = VI

P = 230 × 4

P = 920 W

The appliance transfers electrical energy at a rate of 920 J/s.

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Calculating Current from Power

Rearrange:

P = VI

to obtain:

I = P/V

Suppose a 1200 W appliance operates at 240 V.

I = 1200/240

I = 5 A

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Calculating Voltage from Power

Rearrange:

P = VI

to obtain:

V = P/I

Suppose:

P = 600 W

I = 5 A

Then:

V = 600/5

V = 120 V

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Power in Resistors

For a resistor:

P = VI

Using Ohm's Law:

V = IR

we can derive other power equations.

Substitute V = IR into P = VI:

P = (IR)I

Therefore:

P = I²R

This equation is particularly useful when current and resistance are known.

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Example Using P = I²R

A 20 Ω resistor carries:

I = 3 A

Power:

P = I²R

P = 3² × 20

P = 9 × 20

P = 180 W

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Another Power Equation

Ohm's Law also gives:

I = V/R

Substitute this into:

P = VI

P = V(V/R)

Therefore:

P = V²/R

This equation is useful when voltage and resistance are known.

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Example Using P = V²/R

A resistor has:

R = 50 Ω

and:

V = 100 V

Power:

P = V²/R

P = 100²/50

P = 10,000/50

P = 200 W

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The Three Main Electrical Power Equations

For suitable circuit situations:

P = VI

P = I²R

P = V²/R

Choose the equation based on the quantities provided.

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Choosing the Correct Equation

Suppose a question gives:

V = 12 V

R = 6 Ω

You could first calculate current:

I = V/R

I = 12/6

I = 2 A

Then:

P = VI

P = 12 × 2

P = 24 W

Or calculate directly:

P = V²/R

P = 12²/6

P = 144/6

P = 24 W

Both methods agree.

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Power Ratings

Electrical devices usually have a power rating.

Examples might include:

• LED lamp: 10 W
• television: 100 W
• refrigerator: 150 W while running
• hair dryer: 1800 W
• kettle: 2000 W

The power rating indicates the rate at which the device is designed to transfer energy under its rated operating conditions.

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Reading an Appliance Rating

Suppose an appliance is labelled:

230 V, 1150 W

This tells us:

rated voltage = 230 V

rated power = 1150 W

The expected current can be estimated:

I = P/V

I = 1150/230

I = 5 A

This information can help determine appropriate:

• wiring
• switches
• circuit protection
• power supplies

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What Does a 2000 W Rating Mean?

A 2000 W appliance transfers energy at approximately:

2000 J/s

when operating at its rated conditions.

If it operates for 10 seconds:

E = Pt

E = 2000 × 10

E = 20,000 J

If it operates for 60 seconds:

E = 2000 × 60

E = 120,000 J

or:

120 kJ

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Power and Operating Time

Power alone does not determine total energy consumption.

Energy is:

E = Pt

Therefore, both:

• power
• operating time

must be considered.

A low-power device operating continuously may consume more energy than a high-power device used briefly.

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Comparing Two Appliances

Appliance A:

P = 2000 W

operates for:

5 minutes

Appliance B:

P = 200 W

operates for:

2 hours

Appliance A

5 min = 300 s

E = Pt

E = 2000 × 300

E = 600,000 J

Appliance B

2 h = 7200 s

E = 200 × 7200

E = 1,440,000 J

Although Appliance A has ten times the power, Appliance B transfers more total energy because it operates much longer.

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Power and Energy Consumption

Household electrical energy is commonly measured in:

kilowatt-hours (kWh)

Convert power into kilowatts:

1000 W = 1 kW

Then use:

E = Pt

with:

• P in kW
• t in hours
• E in kWh

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Example in Kilowatt-Hours

A 2 kW heater operates for:

3 h

Energy:

E = Pt

E = 2 × 3

E = 6 kWh

Since:

1 kWh = 3.6 MJ

the energy is also:

6 × 3.6

= 21.6 MJ

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Comparing Appliance Energy Consumption

Consider two appliances.

Kettle

Power:

2.0 kW

Operating time:

0.10 h

Energy:

E = 2.0 × 0.10

E = 0.20 kWh

Television

Power:

0.15 kW

Operating time:

4 h

Energy:

E = 0.15 × 4

E = 0.60 kWh

Although the kettle is much more powerful, the television consumes more energy in this example because it operates for longer.

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Calculating Operating Cost

If electricity is charged per kilowatt-hour:

cost = energy used × price per kWh

Suppose a 1.5 kW heater operates for 4 hours.

Energy:

E = 1.5 × 4

E = 6 kWh

If electricity costs:

$0.30/kWh

then:

cost = 6 × 0.30

cost = $1.80

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Daily and Monthly Energy Use

A 100 W device operates for 8 hours each day.

Convert:

100 W = 0.100 kW

Daily energy:

E = 0.100 × 8

E = 0.80 kWh

For 30 days:

E = 0.80 × 30

E = 24 kWh

If electricity costs $0.30/kWh:

cost = 24 × 0.30

cost = $7.20

Small power ratings can therefore become significant when devices operate for long periods.

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Power and Current

From:

P = VI

for a fixed voltage:

P ∝ I

A higher-power device generally requires more current when operated at the same voltage.

For example, at 230 V:

230 W appliance

I = 230/230

I = 1 A

2300 W appliance

I = 2300/230

I = 10 A

The higher-power appliance requires much more current.

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Why Current Matters

Large currents can cause significant heating in wires.

Heating power is:

P = I²R

If current doubles:

I → 2I

then:

P → (2I)²R

P → 4I²R

Therefore:

doubling the current produces four times the heating power

for the same resistance.

This is why high-current circuits require suitable wiring and protection.

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Electrical Heating

In a resistor:

P = I²R

Electrical energy is transferred mainly into thermal energy.

This is useful in devices such as:

• kettles
• toasters
• electric ovens
• hair dryers
• space heaters

A heating element is designed to transfer electrical energy into thermal energy efficiently for the intended purpose.

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Power Dissipation in a Resistor

Suppose:

R = 10 Ω

I = 0.5 A

Then:

P = I²R

P = 0.5² × 10

P = 0.25 × 10

P = 2.5 W

If the resistor is rated for only:

0.25 W

it would be unsuitable because the expected power dissipation greatly exceeds its rating.

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Resistor Power Ratings

Real resistors have maximum power ratings.

Common ratings include:

• 0.125 W
• 0.25 W
• 0.5 W
• 1 W
• 2 W

Higher-power resistors are generally physically larger because they need to transfer thermal energy to their surroundings safely.

Circuit designers should choose a resistor whose power rating safely exceeds the expected operating power.

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Power in Series Circuits

In a series circuit:

current is the same through each resistor

Since:

P = I²R

a larger resistance dissipates more power when the same current flows through all components.

Suppose:

I = 2 A

R₁ = 2 Ω

R₂ = 5 Ω

Then:

P₁ = 2² × 2

P₁ = 8 W

P₂ = 2² × 5

P₂ = 20 W

The larger resistor transfers energy at a greater rate.

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Power in Parallel Circuits

In a parallel circuit:

voltage is the same across each branch

Since:

P = V²/R

at the same voltage, a lower resistance dissipates more power.

Suppose:

V = 12 V

R₁ = 6 Ω

R₂ = 12 Ω

Then:

P₁ = 12²/6

P₁ = 24 W

P₂ = 12²/12

P₂ = 12 W

The lower-resistance branch transfers energy at the greater rate.

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Total Power in a Circuit

The total power supplied by a source equals the total power transferred by the components in an ideal circuit.

P_total = P₁ + P₂ + P₃ + ...

This follows from conservation of energy.

For example:

P₁ = 20 W

P₂ = 30 W

P₃ = 10 W

Then:

P_total = 20 + 30 + 10

P_total = 60 W

The source must supply approximately 60 W.

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Power Supplied by a Battery

For a battery with emf ε supplying current I:

P_source = εI

If:

ε = 12 V

I = 3 A

then:

P_source = 12 × 3

P_source = 36 W

A real battery may transfer some of this power internally because of internal resistance.

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Internal Power Loss

For a battery with internal resistance r:

P_internal = I²r

Suppose:

I = 3 A

r = 0.20 Ω

Then:

P_internal = 3² × 0.20

P_internal = 9 × 0.20

P_internal = 1.8 W

This energy is mainly transferred into thermal energy inside the battery.

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Useful Power from a Real Battery

If the terminal voltage is V:

P_external = VI

The energy balance is:

P_source = P_external + P_internal

or:

εI = VI + I²r

This connects electrical power with conservation of energy.

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Power and Efficiency

Efficiency can be expressed using power:

efficiency = useful power output / total power input × 100%

Suppose a motor receives:

500 W

and produces:

400 W

of useful mechanical power.

Efficiency:

efficiency = 400/500 × 100%

efficiency = 80%

The remaining:

100 W

is transferred into other forms such as thermal energy and sound.

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Comparing Two Motors

Motor A:

input power = 600 W

useful output = 450 W

Efficiency:

450/600 × 100%

= 75%

Motor B:

input power = 550 W

useful output = 440 W

Efficiency:

440/550 × 100%

= 80%

Motor B uses less input power and has greater efficiency, although Motor A produces slightly more useful output power.

This shows why several factors should be considered when comparing devices.

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Power Ratings and Performance

A higher power rating does not necessarily mean that a device is "better."

It means the device transfers energy at a greater rate under its rated conditions.

Whether this is desirable depends on the purpose.

For example:

• a high-power kettle may heat water faster
• a high-power heater may warm a room faster
• a lower-power lamp may produce similar light if it is more efficient
• a lower-power computer may be preferable if it provides sufficient performance

Power should therefore be interpreted in context.

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Power Ratings and Safety

Power ratings can also help determine electrical safety requirements.

A high-power appliance usually draws a larger current.

For example:

P = 3000 W

V = 230 V

I = P/V

I = 3000/230

I ≈ 13.0 A

Wires, plugs, switches and protective devices must be capable of safely carrying the required current.

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Appliance Labels

Electrical appliance labels may include:

• rated voltage
• rated current
• rated power
• frequency
• safety information

These values help users determine whether the device is suitable for a particular electrical supply.

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Worked Multi-Step Problem

A heater has:

R = 46 Ω

and is connected to:

V = 230 V

Calculate current

I = V/R

I = 230/46

I = 5 A

Calculate power

P = VI

P = 230 × 5

P = 1150 W

Alternatively:

P = V²/R

P = 230²/46

P = 1150 W

Energy transferred in 10 minutes

10 min = 600 s

E = Pt

E = 1150 × 600

E = 690,000 J

or:

690 kJ

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Worked Appliance Cost Problem

A 2.4 kW appliance operates for:

30 minutes per day

for:

20 days

Convert:

30 min = 0.5 h

Daily energy:

E = 2.4 × 0.5

E = 1.2 kWh

Over 20 days:

E = 1.2 × 20

E = 24 kWh

If electricity costs:

$0.28/kWh

cost:

24 × 0.28

= $6.72

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Choosing the Right Power Equation

A useful strategy is to identify what the question gives you.

If you know:

energy and time → P = E/t

voltage and current → P = VI

current and resistance → P = I²R

voltage and resistance → P = V²/R

Do not calculate unnecessary quantities if a direct equation is available.

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Unit Conversions

Power calculations often require unit conversions.

Power

1000 W = 1 kW

1 MW = 1,000,000 W

Current

1000 mA = 1 A

Time

60 s = 1 min

3600 s = 1 h

Energy

1 kWh = 3.6 MJ

Always check the required units before substituting into an equation.

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Example with Milliamps

An electronic component operates at:

V = 12 V

I = 250 mA

Convert:

250 mA = 0.250 A

Then:

P = VI

P = 12 × 0.250

P = 3.0 W

Failing to convert mA to A would produce an answer 1000 times too large.

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Power and Conservation of Energy

Electrical power is ultimately about the rate of energy transfer.

For a circuit:

power supplied = total rate of energy transfer

In an ideal model:

P_source = ΣP_components

In a real system, all energy transfers must still be included.

Some may be:

• useful mechanical power
• useful light
• useful heating
• unwanted heating
• sound

Energy is conserved even when a device is inefficient.

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Common Mistakes

Mistake: Confusing power with energy.
Power is the rate of energy transfer; energy is the total amount transferred.

Mistake: Saying watts measure energy.
Watts measure power. Joules and kilowatt-hours measure energy.

Mistake: Forgetting to square current in P = I²R.
The equation contains I², not I.

Mistake: Forgetting to square voltage in P = V²/R.

Mistake: Using milliamps as though they were amperes.
Convert mA to A.

Mistake: Assuming the highest-power appliance always uses the most energy.
Operating time matters.

Mistake: Assuming a higher power rating always means greater efficiency.
Power and efficiency are different quantities.

Mistake: Ignoring component power ratings.
A resistor may have the correct resistance but still be unable to safely dissipate the required power.

Mistake: Mixing watts and kilowatts in kWh calculations.
Use power in kilowatts when calculating kWh.

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Key Terms

Electrical power — rate at which electrical energy is transferred.

Power — energy transferred per unit time.

Watt (W) — SI unit of power; one joule per second.

Kilowatt (kW) — 1000 watts.

Power rating — power a device is designed to transfer under specified operating conditions.

Power dissipation — rate at which electrical energy is transferred by a component, often as thermal energy.

Joule heating — transfer of electrical energy into thermal energy due to resistance.

Energy consumption — total amount of electrical energy used or transferred over time.

Kilowatt-hour (kWh) — unit of energy equal to 3.6 MJ.

Efficiency — proportion of input power or energy transferred usefully.

Rated voltage — voltage at which a device is designed to operate.

Rated current — current expected under specified operating conditions.

Power loss — power transferred in an unwanted form, commonly thermal energy.

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Key Takeaways

• Electrical power is the rate at which electrical energy is transferred.
• The SI unit of power is the watt.
• 1 W = 1 J/s.
• The basic power equation is P = E/t.
• Electrical power can be calculated using P = VI.
• For resistors, power can also be calculated using P = I²R and P = V²/R.
• The appropriate equation depends on the quantities provided.
• Power ratings describe the rate of energy transfer under specified operating conditions.
• Higher power means faster energy transfer, not necessarily greater total energy consumption.
• Total energy depends on both power and operating time.
• E = Pt.
• Appliance energy consumption is commonly measured in kWh.
• 1 kWh = 3.6 MJ.
• High-power devices generally draw larger currents when operating at the same voltage.
• Heating in resistive components follows P = I²R.
• Doubling current produces four times the resistive heating for the same resistance.
• Components must have suitable power ratings.
• Total circuit power is the sum of the power transferred by its components.
• Real batteries can lose power internally through their internal resistance.
• Efficiency compares useful output power with total input power.
• Appliance comparisons should consider power, operating time, energy consumption, efficiency and useful performance.

Check Your Understanding

1. Define electrical power.
2. What is the SI unit of power?
3. What does a power of 1 W mean?
4. Explain the difference between electrical power and electrical energy.
5. A device transfers 12,000 J in 60 s. Calculate its power.
6. A device operates at 12 V and draws 3 A. Calculate its power.
7. A 230 V appliance is rated at 1840 W. Calculate its current.
8. A 5 Ω resistor carries 4 A. Calculate its power using P = I²R.
9. A 20 Ω resistor has 10 V across it. Calculate its power using P = V²/R.
10. Derive P = I²R from P = VI and Ohm's Law.
11. Derive P = V²/R from P = VI and Ohm's Law.
12. Explain what a 2000 W rating on a heater means.
13. A 2 kW heater operates for 2.5 hours. Calculate the energy consumed in kWh.
14. Convert the answer to Question 13 into joules.
15. A 100 W television operates for 6 hours while a 2000 W kettle operates for 6 minutes. Which device transfers more energy?
16. Explain why doubling current through a resistor causes four times the heating power.
17. A motor receives 800 W and provides 680 W of useful mechanical power. Calculate its efficiency.
18. Explain why a higher-power appliance is not necessarily more energy efficient.
19. A 0.25 W resistor is expected to dissipate 1.5 W. Explain why this is a problem and how the design could be improved.
20. Explain how power rating, operating time, efficiency and energy consumption should all be considered when comparing electrical appliances.