Prepare for University Studies & Career Advancement

Energy and Fields

Course: Prep4Uni Physics 1 Chapter 1: Quantities and Measurement | Chapter 2: Forces and Moments | Chapter 3: Motion and Forces | Chapter 4: Energy and Fields | Chapter 5: Projectile Motion

Overview

This topic introduces the principles of energy, work, and power, along with the concepts of gravitational and electric fields. These ideas help us understand how forces can transfer energy, store it in various forms, and influence the motion of particles over a distance.

Contents

  1. Energy types and transfer
  2. Work, kinetic and potential energy
  3. Fields and Field Lines
  4. Equipotentials and Work in Fields
  5. Power and efficiency

Learning Outcomes

By the end of this section, students should be able to:
  • Identify energy stores and describe energy transfers
  • Apply the principle of conservation of energy
  • Define and calculate work, kinetic energy, and potential energy
  • Understand gravitational and electric fields and interpret field lines
  • Use equipotential concepts and relate work to potential energy changes
  • Calculate power and efficiency and apply these ideas to real-world devices

1. Energy Types and Transfers

The total energy of an isolated system stays constant. Energy can be stored in different forms and transferred from one system to another, but it is never created or destroyed.
Main energy stores (with quick cues):
  • Kinetic — motion; Ek = ½ m v2.
  • Gravitational potential — height in a field; Eg = m g h.
  • Elastic (strain) — deformation; Ee = ½ k x2.
  • Chemical — bonds (fuels, food, batteries).
  • Thermal — temperature/random motion; ΔE = m c ΔT.
  • Electrical — separated charges / current in circuits.
  • Nuclear — binding energy; E = Δm c2.
Energy transfers:
  • By mechanical work (forces doing work), e.g., W = F s.
  • By electrical work, e.g., W = V I t.
  • By heating (conduction, convection, radiation).
  • By waves (e.g., light, sound).
Conservation of energy:
  • Energy cannot be created or destroyed — only transferred or transformed.
Examples of energy transfer:
  • Falling object: gravitational potential → kinetic (ignore air resistance).
  • Rubber band: elastic → kinetic.
  • Battery torch: chemical → electrical work → light + thermal.
  • Kettle: electrical → heating → thermal energy of water.
  • Speaker: electrical → sound waves → kinetic/thermal of air.
Learning-Check 1
Q: Name two ways energy can be transferred from one system to another.Answer: By mechanical work and by heating.
Learning-Check 2
Q: What type of energy store increases when a spring is compressed?Answer: Elastic potential energy.

2. Work, Kinetic Energy, and Potential Energy (unit: joule, J)

Work done (W): W = F × d. The figure shows a block pushed by a force F through a displacement d.
Block pushed along a horizontal surface: force F acts in the same direction as displacement d.
Work: W = F × d
Kinetic energy (KE): KE = ½mv2.
A mass m moving with velocity v.
Moving mass: KE = ½mv2
Elastic potential energy (EPE): EPE = ½kx2. For a spring obeying Hooke’s law (F = kx), the shaded triangular area under the F–x graph equals the stored elastic energy.
Force–extension graph of a spring; triangular area under the straight line equals elastic potential energy.
Elastic energy from force–extension graph: ½kx2
Gravitational potential energy (GPE): GPE = mgh (or ΔEp = mgΔh).
Raising a block through height h increases gravitational potential energy mgh.
Raise by height h: GPE = mgh
Worked examples
  • Work: A 10 N force moves an object 4 m. W = 10 × 4 = 40 J.
  • Kinetic energy: 2 kg at 3 m/s: KE = ½ × 2 × 32 = 9 J.
  • Elastic energy: k = 200 N/m, x = 0.1 m: EPE = ½ × 200 × (0.1)2 = 1 J.
  • Gravitational energy: 5 kg lifted 3 m: GPE = 5 × 9.8 × 3 = 147 J.
Learning-Check 1
Q: A 10 N force moves an object 4 m in the direction of the force. How much work is done?Answer: W = F d = 10 × 4 = 40 J.
Learning-Check 2
Q: What is the kinetic energy of a 0.5 kg ball moving at 6 m/s?Answer: KE = ½mv2 = 0.5 × 0.5 × 36 = 9 J.

3. Fields and Field Lines

Gravitational field. A region in which a mass experiences a gravitational force.
  • Field strength: g = F / m (near Earth, g ≈ 9.8 m s-2, directed toward Earth’s centre).
  • Field lines: point toward the centre of mass; lines never cross.
Gravitational field lines pointing radially inward toward Earth. Field strength g = F/m.
Gravitational field: lines are radial and inward.
Electric field. A region in which a charge experiences an electric force.
  • Field strength: E = F / q.
  • Direction: lines start on + charges and end on −; they point away from + and toward −.
  • Uniform field (parallel plates): approximately E = V / d — straight, parallel, equally spaced lines.
Electric field diagram: lines leaving positive charges and entering negative charges; uniform between parallel plates.
Electric field patterns and a uniform field between parallel plates.
What field lines show:
  • Direction of the force on a small test mass/charge placed at that point.
  • Strength — denser (closer) lines indicate a stronger field.
  • In electrostatics, lines meet conducting surfaces at right angles; lines do not intersect.
Learning-Check 1
Q: In a gravitational field, in what direction do the field lines point?Answer: Toward the mass (e.g., toward Earth’s centre).
Learning-Check 2
Q: Are electric field lines closer together in strong or weak fields?Answer: Closer together in stronger fields (larger E).

4. Equipotentials and Work in Fields

Equipotential lines/surfaces. Sets of points that share the same potential (V for electric fields; potential energy per unit charge/mass). Moving along an equipotential does no work:
  • ΔW = 0 and ΔV = 0 along an equipotential.
  • Equipotentials are always perpendicular to field lines.
  • Closer spacing of equipotentials ⇒ stronger field (larger gradient).
Equipotential lines drawn perpendicular to field lines
Equipotentials: equal potential everywhere along a line/curve.
Work and potential energy:
  • Gravitational: ΔEp = m g Δh. If the object moves down (Δh < 0), gravitational potential energy decreases and the same amount can appear as kinetic energy (neglecting losses).
  • Electric: W = q ΔV. For a uniform field between plates, ΔV = E d and W = q E d.
Work from potential difference and the relation between E, d, and ΔV
Work from potential differences: W = q ΔV; in a uniform field, ΔV = E d.
Worked example

A 2 kg mass descends 5 m. ΔEp = m g Δh = 2 × 9.8 × 5 = 98 J. This 98 J is released (e.g., as kinetic energy) if non-conservative losses are negligible.

Learning-Check 1
Q: True or False: Work is done when a charge moves along an equipotential line.Answer: False. Along an equipotential, ΔV = 0 ⇒ W = q ΔV = 0.
Learning-Check 2
Q: A 3 C charge moves through a potential difference of 12 V. How much work is done?Answer: W = q ΔV = 3 × 12 = 36 J.

5. Power and Efficiency

Power, P — rate of doing work / transferring energy.
  • P = W / t = ΔE / t (unit: watt, W = J·s-1).
  • For constant speed in a straight line: P = F v.
Efficiency — fraction of input energy (or power) that becomes useful output.
  • Efficiency = (useful output / total input) × 100%.
  • Equivalently with power: Efficiency = (Puseful / Pin) × 100%.
Energy flow diagram for a motor showing useful output versus wasted energy
Energy flow: only part of the input becomes useful output; the rest is dissipated (e.g., heat, sound).
Worked example

A motor delivers 400 J of useful energy from 500 J input. Efficiency: (400 / 500) × 100% = 80%.

Learning-Check 1
Q: A 60 W bulb operates for 10 s. How much energy does it use?Answer: E = P t = 60 × 10 = 600 J.
Learning-Check 2
Q: A machine uses 1000 J of energy but delivers 300 J of useful work. What is its efficiency?Answer: (300 / 1000) × 100% = 30%.

Key Concepts Recap

ConceptFormulaNotes
WorkW = F d cos(θ)θ is angle between force and displacement
Kinetic energyKE = ½ m v2Always positive
Gravitational potentialGPE = m g hRelative to a reference height
PowerP = W / t or P = F vRate of doing work
Efficiency(useful / total) × 100%Expressed as a percentage

Interactive Simulator: Work, Energy & Field Energy Calculator

Energy & Field Trajectory Simulator

Adjust parameters to calculate work done, energy transfers, and visualize displacement over time with explicit axes.

Results:

Interactive Review Questions

Review Question 1: Why is no work done moving along an equipotential surface?

Along an equipotential surface, the potential energy difference (ΔV) is zero. Since work done in a field is given by W = qΔV or mΔV, zero potential change directly results in zero work performed.

Review Question 2: How does the spacing of field lines relate to field strength?

Field strength is proportional to line density. Closer, tightly spaced field lines represent a strong field region (high field magnitude), while spread-out lines indicate a weaker field region.

Review Question 3: What causes energy dissipation in mechanical systems?

Non-conservative forces such as friction, air resistance, and mechanical drag transform useful mechanical energy (kinetic or potential) into non-recoverable thermal energy and acoustic energy, reducing overall system efficiency.

Frequently Asked Questions

Can kinetic energy ever be negative?

No. Kinetic energy is defined as KE = ½mv2. Mass m is positive, and velocity squared v2 is always non-negative. Therefore, kinetic energy is always zero or positive.

What is the angle between equipotential lines and electric field lines?

Equipotential lines and field lines are always mutually perpendicular (90 degrees). If field lines had a component along an equipotential, work would be required to move along it, contradicting the definition of an equipotential.

Why can efficiency never exceed 100%?

Due to the Law of Conservation of Energy, energy cannot be created from nothing. The useful energy output can never exceed the total energy input, making efficiencies over 100% physically impossible.

Consolidated Cluster Navigation

Classical Mechanics Sub-Hub

Explore fundamental physical dynamics and systemic mechanics principles across structured physics pathways. Visit Classical Mechanics

Dynamics and Kinematics

Master the quantitative relationships linking motion, accelerations, and field-based forces. Visit Dynamics & Kinematics

Fundamental Forces of Nature

Understand the primary field interactions governing gravitational, electromagnetic, and subatomic forces. Visit Fundamental Forces

Next Topic: Projectile Motion

Advance to Chapter 5 to analyze two-dimensional parabolic trajectories under uniform gravitational field acceleration. Proceed to Projectile Motion

Practice Module & Exercises

Section 1: Foundational Concept Review Questions (8 Questions)

  1. What is meant by an energy “store”?Answer: A way energy is held in a system (e.g., chemical, thermal, gravitational).
  2. Give two examples of energy-transfer mechanisms.Answer: Mechanical work (force × distance) and heating (conduction/convection/radiation).
  3. State the principle of conservation of energy.Answer: Energy cannot be created or destroyed—only transferred or transformed.
  4. In a swinging pendulum, which stores interchange?Answer: Gravitational potential ↔ kinetic.
  5. Distinguish an open system from a closed system in energy terms.Answer: Open system: exchanges energy with surroundings; Closed system: does not exchange energy with surroundings.
  6. Define “work done” by a force.Answer: W = F s cos(θ) (work performed in the direction of force displacement).
  7. What is the SI unit of work and energy?Answer: Joule (J).
  8. State the work–energy theorem.Answer: The net work done on an object equals its change in kinetic energy.

Section 2: Analytical & Scenario-Based Questions (8 Questions)

  1. Explain why moving a charge along an equipotential line requires zero work.Answer: Potential difference ΔV along an equipotential is zero. Since W = qΔV, the total work executed is zero.
  2. How do field lines visually indicate the strength of an electric or gravitational field?Answer: The density of the field lines represents field strength; closer lines indicate a stronger field magnitude.
  3. Describe the direction of electric field lines between two parallel oppositely charged plates.Answer: Field lines are straight, parallel, equally spaced, and directed perpendicular from the positive plate to the negative plate.
  4. Why does an ideal spring's force-extension graph form a triangle whose area equals stored energy?Answer: Force increases linearly with extension (F = kx); the integral/area under the linear slope gives work done: ½kx2.
  5. How does non-conservative friction affect energy transformation in a falling object?Answer: Air resistance does negative work, converting part of the initial gravitational potential energy into thermal energy instead of pure kinetic energy.
  6. What happens to net gravitational field strength at the exact midpoint between two equal mass spheres?Answer: The gravitational field vectors are equal in magnitude but opposite in direction, so they cancel, yielding a net field strength of zero.
  7. Explain why power can be expressed as P = Fv for constant velocity motion.Answer: Since P = W / t = (F × s) / t and s / t = v, substituting speed yields P = Fv.
  8. Why can an energy transfer process never achieve greater than 100% efficiency?Answer: Creating extra energy violates the first law of thermodynamics (conservation of energy); total useful output cannot exceed total input.

Section 3: Numerical Problems & Step-by-Step Solutions (8 Problems)

  1. A 2 kg mass accelerates from rest to 5 m s-1. Calculate its final kinetic energy.Solution: KE = ½ m v2 = ½ × 2 × 52 = 25 J.
  2. A 1 kg object is lifted 3 m vertically near Earth's surface. Calculate the gain in potential energy.Solution: ΔEp = m g Δh = 1 × 9.8 × 3 = 29.4 J.
  3. A spring with spring constant k = 800 N m-1 is compressed by 0.10 m. Find the elastic potential energy stored.Solution: Eelastic = ½ k x2 = ½ × 800 × (0.10)2 = 4 J.
  4. A 3 C electric charge moves through a potential difference of 12 V. Calculate the work done.Solution: W = q ΔV = 3 × 12 = 36 J.
  5. Two parallel plates are separated by 0.02 m with a voltage of 100 V. Determine the electric field strength.Solution: E = V / d = 100 / 0.02 = 5.0 × 103 N C-1.
  6. An elevator weighing 1000 kg ascends 5 m in 10 s at constant speed. Compute the required power output.Solution: W = m g h = 1000 × 9.8 × 5 = 49,000 J. P = W / t = 49,000 / 10 = 4900 W (4.9 kW).
  7. A motor delivers 400 J of useful mechanical work from an input energy of 500 J. Calculate its efficiency.Solution: Efficiency = (400 / 500) × 100% = 80%.
  8. A pump delivers a volume flow rate Q = 0.05 m3 s-1 to a height h = 10 m using input power Pin = 10 kW. Calculate pump efficiency (ρ = 1000 kg m-3).Solution: Mass flow rate m/t = ρ × Q = 1000 × 0.05 = 50 kg s-1. Useful power Puseful = (m/t) g h = 50 × 9.8 × 10 = 4900 W = 4.9 kW. Efficiency = (4.9 / 10) × 100% = 49%.
Last updated: 05 Aug 2026