Introduction
Power formulas are fundamental equations used across various fields of science and engineering. This comprehensive guide covers all essential power formulas from basic physics to advanced electrical engineering, providing students with a complete reference for academic success.
Basic Power Formulas (Mechanical Physics)
| Formula | Equation | Variables | Units | Application |
|---|---|---|---|---|
| Basic Power | P = W/t | P = Power, W = Work, t = time | Watts (W) | General power calculation |
| Power from Force | P = F × v | P = Power, F = Force, v = velocity | Watts (W) | Moving objects under force |
| Rotational Power | P = τ × ω | P = Power, τ = torque, ω = angular velocity | Watts (W) | Rotating machinery |
| Power from Energy | P = E/t | P = Power, E = Energy, t = time | Watts (W) | Energy conversion rates |
Electrical Power Formulas
| Formula | Equation | Variables | Units | Application |
|---|---|---|---|---|
| Basic Electrical Power | P = V × I | P = Power, V = Voltage, I = Current | Watts (W) | DC circuits |
| Power with Resistance | P = I²R | P = Power, I = Current, R = Resistance | Watts (W) | Resistive circuits |
| Power with Voltage | P = V²/R | P = Power, V = Voltage, R = Resistance | Watts (W) | When current unknown |
| AC Power (Real) | P = V × I × cos φ | P = Power, V = RMS Voltage, I = RMS Current, φ = phase angle | Watts (W) | AC circuits |
| Power Dissipation | P = E²/R | P = Power, E = EMF, R = Resistance | Watts (W) | Heat dissipation |
AC Power System Formulas
| Formula | Equation | Variables | Units | Application |
|---|---|---|---|---|
| Apparent Power | S = V × I | S = Apparent power, V = RMS Voltage, I = RMS Current | VA (Volt-Amperes) | Total power in AC circuits |
| Reactive Power | Q = V × I × sin φ | Q = Reactive power, φ = phase angle | VAR (Volt-Amperes Reactive) | Non-working power |
| Power Triangle | S² = P² + Q² | S = Apparent, P = Real, Q = Reactive power | Various | Power relationships |
| Power Factor | pf = cos φ = P/S | pf = Power factor, P = Real power, S = Apparent power | Dimensionless (0-1) | Efficiency indicator |
Three-Phase Power Formulas
| Formula | Equation | Variables | Units | Application |
|---|---|---|---|---|
| 3-Phase Real Power | P = √3 × VL × IL × cos φ | VL = Line voltage, IL = Line current | Watts (W) | Balanced 3-phase systems |
| 3-Phase Apparent Power | S = √3 × VL × IL | VL = Line voltage, IL = Line current | VA | Total 3-phase power |
| 3-Phase Reactive Power | Q = √3 × VL × IL × sin φ | φ = phase angle | VAR | 3-phase reactive power |
| Star Connection Power | P = 3 × Vph × Iph × cos φ | Vph = Phase voltage, Iph = Phase current | Watts (W) | Star-connected loads |
| Delta Connection Power | P = 3 × VL × IL × cos φ / √3 | VL = Line voltage, IL = Line current | Watts (W) | Delta-connected loads |
Power in Optics (Lens Power)
| Formula | Equation | Variables | Units | Application |
|---|---|---|---|---|
| Lens Power | P = 1/f | P = Power of lens, f = focal length | Diopters (D) | Lens strength |
| Lens Power (Refractive Index) | P = (n-1)/R | n = refractive index, R = radius of curvature | Diopters (D) | Lens design |
| Combined Lens Power | P = P₁ + P₂ | P₁, P₂ = individual lens powers | Diopters (D) | Multiple lens systems |
| Lens Maker’s Formula Power | P = (n-1)(1/R₁ – 1/R₂) | R₁, R₂ = radii of curvature | Diopters (D) | Lens manufacturing |
Mathematical Powers and Exponents
| Formula | Rule | Example | Application |
|---|---|---|---|
| Product Rule | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2² = 2⁵ | Multiplying same bases |
| Quotient Rule | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 5⁴ ÷ 5² = 5² | Dividing same bases |
| Power Rule | (aᵐ)ⁿ = aᵐⁿ | (3²)³ = 3⁶ | Power of a power |
| Power of Product | (ab)ⁿ = aⁿbⁿ | (2×3)² = 2²×3² | Product raised to power |
| Power of Quotient | (a/b)ⁿ = aⁿ/bⁿ | (4/2)³ = 4³/2³ | Quotient raised to power |
| Zero Power | a⁰ = 1 | 7⁰ = 1 | Any number to power zero |
| Negative Power | a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/2³ | Negative exponents |
Dimensional Formula of Power
| Quantity | Dimensional Formula | Derivation | SI Unit |
|---|---|---|---|
| Power | [M L² T⁻³] | P = Work/Time = [M L² T⁻²]/[T] | Watt (kg⋅m²⋅s⁻³) |
| Electrical Power | [M L² T⁻³] | P = V × I = [M L² T⁻³ I⁻¹] × [I] | Watt |
| Mechanical Power | [M L² T⁻³] | P = F × v = [M L T⁻²] × [L T⁻¹] | Watt |
Specialized Power Formulas
| Formula | Equation | Variables | Units | Application |
|---|---|---|---|---|
| Radiated Power | P = σ × A × T⁴ | σ = Stefan-Boltzmann constant, A = area, T = temperature | Watts (W) | Heat radiation |
| Sound Power | P = I × A | I = Sound intensity, A = Area | Watts (W) | Acoustics |
| Solar Power | P = G × A × η | G = Solar irradiance, A = Area, η = efficiency | Watts (W) | Solar panels |
| Wind Power | P = ½ × ρ × A × v³ × Cp | ρ = air density, A = area, v = wind speed, Cp = power coefficient | Watts (W) | Wind turbines |
| Hydraulic Power | P = ρ × g × Q × H | ρ = fluid density, g = gravity, Q = flow rate, H = head | Watts (W) | Water turbines |
Power Conversion Formulas
| Conversion | Formula | Notes |
|---|---|---|
| Watts to Horsepower | HP = W / 745.7 | Mechanical horsepower |
| Watts to BTU/hr | BTU/hr = W × 3.412 | Heat equivalent |
| kW to MW | MW = kW / 1000 | Electrical power scaling |
| dBm to Watts | W = 10^((dBm-30)/10) | Logarithmic power scale |
Main Concepts and Applications
Power Factor Importance
- Definition: Ratio of real power to apparent power
- Range: 0 to 1 (or 0% to 100%)
- Impact: Higher power factor means better efficiency
- Improvement: Use capacitors for inductive loads
Three-Phase Advantages
- Efficiency: More efficient power transmission
- Balance: Better load distribution
- Economy: Less conductor material needed
- Applications: Industrial motors, power distribution
Lens Power Applications
- Eyeglasses: Correcting vision problems
- Cameras: Focusing light for image formation
- Telescopes: Magnifying distant objects
- Microscopes: Magnifying small objects
Study Tips for Students
- Understand Units: Always check dimensional consistency
- Practice Problems: Work through numerical examples
- Draw Diagrams: Visualize power flow and relationships
- Connect Concepts: Link formulas to physical principles
- Use Mnemonics: “PIE” for Power = I²R, P = I×V, P = V²/R
Common Mistakes to Avoid
- Confusing RMS and peak values in AC calculations
- Forgetting phase angle in AC power formulas
- Mixing up lens power with electrical power
- Incorrect unit conversions
- Not considering power factor in efficiency calculations
Conclusion
This comprehensive guide covers all essential power formulas across multiple disciplines. Regular practice with these formulas will build strong problem-solving skills and deepen understanding of power concepts in physics, electrical engineering, optics, and mathematics. Remember to always verify units and understand the physical meaning behind each formula.
Frequently Asked Questions (FAQs) on Power Formulas
Q. What is the basic formula for power?
The basic power formula is P = W/t, where P is power in watts, W is work done in joules, and t is time in seconds. For electrical circuits, power is calculated as P = V × I (voltage times current) or P = I²R (current squared times resistance).
Q. What is the power factor formula and why is it important?
Answer: The power factor formula is pf = cos φ = P/S, where P is real power (watts) and S is apparent power (VA). Power factor ranges from 0 to 1, indicating how efficiently electrical power is being used. A power factor of 1 (or 100%) means all supplied power is being used effectively, while lower values indicate wasted energy.
Q. How do you calculate 3-phase power?
For balanced three-phase systems, use P = √3 × VL × IL × cos φ, where VL is line voltage, IL is line current, and cos φ is the power factor. This formula gives real power in watts. For apparent power, use S = √3 × VL × IL (in VA).
Q. What is the dimensional formula of power?
The dimensional formula of power is [M L² T⁻³], where M represents mass, L represents length, and T represents time. This can be derived from P = Work/Time = [M L² T⁻²]/[T] = [M L² T⁻³]. In SI units, power is measured in watts (kg⋅m²⋅s⁻³).
Q. What is the difference between real power, apparent power, and reactive power?
- Real Power (P): Actual power consumed, measured in watts (W), calculated as P = V × I × cos φ
- Apparent Power (S): Total power supplied, measured in volt-amperes (VA), calculated as S = V × I
- Reactive Power (Q): Non-working power, measured in VAR, calculated as Q = V × I × sin φ
They relate through the power triangle: S² = P² + Q²
Q. What is the power of lens formula?
The power of lens formula is P = 1/f, where P is power in diopters (D) and f is focal length in meters. A lens with focal length 0.5 m has power of 2 D. Positive power indicates a converging (convex) lens, while negative power indicates a diverging (concave) lens.
Q. How do you calculate power dissipation in a resistor?
Power dissipation in a resistor can be calculated using three formulas:
- P = I²R (when current is known)
- P = V²/R (when voltage is known)
- P = V × I (when both voltage and current are known)
All give power in watts representing heat energy dissipated.
Q. What are the units of power and their conversions?
- SI Unit: Watt (W) = 1 Joule/second
- Horsepower: 1 HP = 745.7 W
- Kilowatt: 1 kW = 1000 W
- Megawatt: 1 MW = 1,000,000 W
- BTU/hour: 1 W = 3.412 BTU/hr
Q. What is the formula for AC power and how does it differ from DC power?
For AC power, P = V × I × cos φ, where cos φ is the power factor accounting for phase difference between voltage and current. DC power is simply P = V × I since there’s no phase difference. AC circuits also have apparent power (S) and reactive power (Q), which don’t exist in DC circuits.
Q. How do you calculate power consumption and electricity cost?
- Calculate power: P (in kW) = V × I × pf / 1000
- Calculate energy: E (in kWh) = P × t (time in hours)
- Calculate cost: Cost = E × rate per kWh
Example: A 1000W appliance running for 5 hours = 5 kWh. At $0.12/kWh, cost = $0.60.
Q. What are the laws of exponents for powers in mathematics?
laws of exponents:
- Product Rule: aᵐ × aⁿ = aᵐ⁺ⁿ
- Quotient Rule: aᵐ ÷ aⁿ = aᵐ⁻ⁿ
- Power Rule: (aᵐ)ⁿ = aᵐⁿ
- Zero Exponent: a⁰ = 1
- Negative Exponent: a⁻ⁿ = 1/aⁿ
Q. What is the relationship between power, energy, and time?
Power is the rate of energy transfer or consumption: P = E/t. Power (watts) equals energy (joules) divided by time (seconds). Higher power means faster energy transfer. A 100W bulb uses 100 joules per second, while a 60W bulb uses 60 joules per second.
Q. How do you improve power factor in electrical systems?
Power factor can be improved by:
- Installing capacitor banks to offset inductive loads
- Using synchronous condensers
- Avoiding operation of equipment at low loads
- Replacing old, inefficient motors
- Better power factor reduces electricity costs and improves system efficiency
Q. What is the formula for power in rotational motion?
For rotational power, P = τ × ω, where τ (tau) is torque in Newton-meters and ω (omega) is angular velocity in radians per second. This is equivalent to P = F × v for linear motion but applied to rotating systems like motors and turbines.
Q. Can power be negative? What does it mean?
Yes, power can be negative, indicating the direction of energy flow. Negative power means a device is supplying or generating power rather than consuming it. For example, a battery being charged has negative power (receiving energy), while discharging has positive power (delivering energy).
Q. What is the formula for solar panel power output?
Solar panel power is calculated as P = G × A × η, where:
- G = Solar irradiance (W/m²), typically 1000 W/m² under standard conditions
- A = Panel area (m²)
- η = Panel efficiency (typically 15-22%)
Example: A 2 m² panel with 20% efficiency: P = 1000 × 2 × 0.20 = 400 W
Q. How do you derive the dimensional formula of power?
Power = Work/Time
- Work = Force × Distance = [M L T⁻²] × [L] = [M L² T⁻²]
- Power = [M L² T⁻²] / [T] = [M L² T⁻³]
This shows power has dimensions of mass times length squared per time cubed.
Q. What is the difference between horsepower and kilowatt?
Both measure power but use different units:
- 1 Horsepower (HP) = 745.7 watts or 0.7457 kW
- 1 Kilowatt (kW) = 1000 watts or 1.34 HP
Horsepower is commonly used for engines and motors (especially in USA), while kilowatts are standard in electrical systems worldwide.
Q. What formulas are used for calculating power loss in transmission lines?
Power loss in transmission lines:
- P_loss = I²R (resistive losses)
- P_loss = (P²R)/V² (when load power and voltage are known)
To minimize losses: increase voltage (reduces current for same power), use thicker conductors (reduces resistance), or reduce transmission distance.
Q. How is average power different from instantaneous power?
Instantaneous Power: Power at a specific moment, p(t) = v(t) × i(t)
- Average Power: Power averaged over a complete cycle, P_avg = (1/T) ∫ p(t) dt
For AC circuits with sinusoidal waveforms, average power = V_rms × I_rms × cos φ, where RMS values are used.