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Energy in Systems and Electric Power

A system is the set of objects you choose to study. Its total energy stays the same unless energy crosses the system boundary as work, heat or radiation: ΔE_system = W + Q. Inside, energy moves between stores such as kinetic, gravitational, elastic, thermal, chemical and electric. Power is the rate of energy transfer, P = ΔE/Δt, in watts. In a circuit the electric power is P = IV; for a resistor this also equals I²R and V²/R. Real devices waste part of the input as heat, so efficiency η = useful output ÷ input is always less than 100%. Designing a device means choosing the input store, the output store and cutting the waste.

🎬 Step-by-step story

  1. Draw a box around what you study. That is the system. Energy only changes when it crosses the box as work or heat.
  2. The motor lifts the crate. The crate gains potential energy mgh. This energy came from the battery's chemical store.
  3. Electric power is P = V × I. Double the voltage with the same current and the power doubles.
  4. Current in the motor coil heats it: P = I²R. This heat is energy that leaves without doing useful work.
  5. Efficiency = useful ÷ input. Useful plus wasted always equals the input. No energy disappears.
  6. Free play: change V, I, efficiency and mass. Predict the lift speed, then check.

Tip: drag the 3D scene to turn it. Use two fingers to zoom.

🤔 Common doubts, cleared

Why does the choice of system change the answer?

If the crate alone is the system, the rope does work on it from outside. If the box includes battery, motor and crate, energy just moves between stores inside. Same physics, different bookkeeping.

Where did the crate's energy actually come from?

From the chemical store in the battery, through the electric current, into the motor and up the rope.

Is P = I²R a different law from P = IV?

No. It is the same formula with V replaced by IR. Use whichever fits the quantities you know.

Why does the motor get warm even when it works well?

Its copper coil has resistance, so current makes heat at the rate I²R. You can see the red glow in step 4.

If energy is conserved, why do we say energy is wasted?

The wasted energy still exists, but it spreads out as heat and we cannot use it easily. The bars in step 5 always add up to the input.

Why does a heavier crate rise more slowly?

Useful power is fixed, and P = mgv, so a bigger m means a smaller v. Try it in free play.

Systems and energy accounting

A system is the group of objects you choose to study. Everything else is the surroundings. The line between them is the boundary.

Energy is never made or destroyed. So the energy of a system changes only if energy crosses the boundary:

ΔEsystem = W + Q (work done on the system plus heat put into it).

Energy stores at the big (macro) scale

All of these are really either motion of particles or energy stored in fields between them. A stretched spring stores energy in the electric fields between its atoms.

Choosing the system matters

If the system is only the crate, the rope does work on it. If the system is crate + Earth, the energy is stored as gravitational potential energy inside the system.

Electric power: P = IV, I²R and V²/R

Power is how fast energy is transferred: P = E ÷ t. Unit: watt (W) = joule per second.

Voltage V is energy per coulomb. Current I is coulombs per second. Multiply them: joules per second. So P = IV.

For a resistor, V = IR. Put this in:

Energy used in time t: E = Pt = IVt. Electricity bills use the kilowatt-hour: 1 kWh = 1000 W × 3600 s = 3.6 × 10⁶ J.

Power in a whole circuit

The battery supplies power εI. This equals the total power used by all resistors (including the battery's internal resistance r): εI = I²R + I²r.

Efficiency and wasted energy

Efficiency η = useful energy out ÷ total energy in (or useful power ÷ input power). Multiply by 100 for a percentage.

No real device reaches 100%. Friction, air drag, sound and the heating of wires (I²R) always take some energy. This wasted energy usually ends up as thermal energy that spreads into the surroundings.

Typical values: electric motor 70–95%, LED bulb about 40%, filament bulb about 5%, car petrol engine about 25–30%, solar panel about 20%.

Designing an energy-conversion device

Engineers follow simple steps:

  1. Name the input store and the output store (e.g. chemical → electric in a battery, light → electric in a solar cell, kinetic → electric in a wind turbine).
  2. Draw the chain of transfers (e.g. falling water → turbine → generator → wires).
  3. Find where energy is wasted and reduce it (oil bearings, thicker wires, insulation).
  4. Check limits: cost, safety, materials, environment.

Try it: make a rubber-band car or a hand-cranked torch. Name every energy transfer and guess where energy is lost. In the 3D, set η to 30% and then 90%, and watch the heat bar shrink.

Key formulas and definitions

Worked examples

1. A 12 V battery drives 2 A through a motor. Find the input power.

P = IV = 2 × 12 = 24 W.

2. A 6 Ω heater carries 3 A. How much power turns into heat?

P = I²R = 3² × 6 = 9 × 6 = 54 W.

3. A 230 V supply is connected across a 46 Ω element. Find the power.

P = V²/R = 230² ÷ 46 = 52,900 ÷ 46 = 1,150 W.

4. A 2 kW geyser runs 30 minutes a day for 30 days. Find the energy in kWh and the cost at ₹8 per kWh.

E per day = 2 kW × 0.5 h = 1 kWh. For 30 days: 30 kWh. Cost = 30 × 8 = ₹240.

5. A motor takes 24 W and lifts a 2 kg crate at 0.73 m/s. Find its efficiency (g = 9.8 m/s²).

Useful power = mgv = 2 × 9.8 × 0.73 ≈ 14.3 W. η = 14.3 ÷ 24 ≈ 0.60 = 60%.

6. A 9 V battery with internal resistance 0.5 Ω drives current through a 4 Ω resistor. Find the power in the resistor and the power wasted inside the battery.

I = ε ÷ (R + r) = 9 ÷ 4.5 = 2 A. In resistor: I²R = 4 × 4 = 16 W. In battery: I²r = 4 × 0.5 = 2 W. Total = 18 W = εI = 9 × 2. ✓

Common mistakes

Practice quiz

1. The unit of power is:
2. A device takes 5 A at 10 V. Its power is:
3. The energy of a system changes only when:
4. Doubling the current through a fixed resistor makes the heat power:
5. A motor uses 100 J and gives 75 J of useful work. Its efficiency is:

Practice: answer these yourself

Type or choose your answer, then press Check. Use a hint if you are stuck; the full solution appears after you answer.

Frequently asked questions

What is the difference between energy and power?

Energy is the amount transferred (joules). Power is how fast it is transferred (watts = joules per second).

Which formula should I use: IV, I²R or V²/R?

All are equal for a resistor. Use I²R when current is given (series), V²/R when voltage is given (parallel), IV for any device.

Why are power lines at high voltage?

For the same power, high voltage means low current, and the heat lost in the wires is I²R, so it falls a lot.

Where this is taught

USA (Common Core, NGSS, AP)Grade 12Electric Circuits
USA (Common Core, NGSS, AP)Grade 12Electric Circuits
USA (Common Core, NGSS, AP)Grade 12Energy

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