D. Gravitational, Electric, and Magnetic Fields · Gravitational, Electric and Magnetic FieldsComparing Fields and the Fundamental Forces
D3.2 D3.1 D2.1About 70 minutes
Learning goals- I can compare and contrast gravitational, electric and magnetic fields: their sources, shapes, strengths and the forces they produce.
- I can use the parallels between Newton's law and Coulomb's law (mass ↔ charge, G ↔ k) to move between the two.
- I can name the four fundamental forces and the exchange particle linked to each.
- I can describe, in general terms, how general relativity and the Standard Model explain forces, and where they disagree.
Words to know
fundamental force
One of the four basic interactions in nature: gravitational, electromagnetic, strong nuclear and weak nuclear. All other forces (friction, tension, normal force) come from these.
exchange particle
In the Standard Model, a particle that carries a force between other particles: the photon (electromagnetic), gluons (strong), W and Z bosons (weak). A graviton has been proposed for gravity but never detected.
Standard Model
The tested theory of particle physics describing the electromagnetic, weak and strong forces and the particles they act on. It does not include gravity.
general relativity
Einstein's 1915 theory of gravity, in which mass and energy curve spacetime and objects follow the straightest possible paths through that curved spacetime.
gravitational wave
A ripple in spacetime that travels at the speed of light, produced by accelerating masses such as two black holes orbiting each other. First detected directly by LIGO in 2015.
inverse-square law
A rule in which a quantity falls off as 1/r². The gravitational field of a mass and the electric field of a point charge both follow it.
Three fields, one family resemblance
Over this unit you have met three fields. Put the equations side by side and the similarity is hard to miss:
- Newton: F = Gm₁m₂/r²
- Coulomb: F = kq₁q₂/r²
Same shape, different "source" (mass or charge) and a very different constant. Magnetism looks different again: its field comes from moving charges and pushes sideways on other moving charges.
This lesson compares the three fields directly (D3.2), then steps back to ask a bigger question (D3.1): how do physicists think forces actually work?
Gravitational and electric fields: a dictionary
You can translate almost any gravity result into an electric one by swapping mass for charge and G for k. The table below is your translation dictionary.
| Idea | Gravitational | Electric |
|---|
| source | mass m (always positive) | charge q (positive or negative) |
| force law | F = Gm₁m₂/r² | F = kq₁q₂/r² |
| constant | G = 6.67 × 10⁻¹¹ N·m²/kg² | k = 8.99 × 10⁹ N·m²/C² |
| field strength | g = F/m = GM/r² (N/kg) | E = F/q = kQ/r² (N/C) |
| potential energy | Eg = −GMm/r (always negative) | Ee = kQq/r (sign depends on the charges) |
| potential | −GM/r (J/kg) | V = kQ/r (J/C, or volts) |
| attract or repel? | always attracts | like charges repel, unlike attract |
| can it be blocked? | no known way to shield | yes: a conducting shell (Faraday cage) |
| uniform version | near Earth's surface (g ≈ 9.8 N/kg) | between parallel plates (E = ΔV/d) |
Worked exampleExample 1: How weak is gravity?
In a hydrogen atom, the electron and proton are about 5.29 × 10⁻¹¹ m apart. Find the ratio of the electric force to the gravitational force between them.
- Electric: Fe = ke²/r². Gravitational: Fg = Gm_e m_p/r².
- The ratio is Fe/Fg = ke²/(Gm_e m_p). The r² cancels, so the distance does not matter.
- ke² = (8.99 × 10⁹)(1.60 × 10⁻¹⁹)² = 2.301 × 10⁻²⁸ N·m².
- Gm_e m_p = (6.67 × 10⁻¹¹)(9.11 × 10⁻³¹)(1.67 × 10⁻²⁷) = 1.015 × 10⁻⁶⁷ N·m².
- Fe/Fg = 2.301 × 10⁻²⁸ / 1.015 × 10⁻⁶⁷ = 2.27 × 10³⁹.
The electric force is about 2.3 × 10³⁹ times stronger, at every distance. Inside atoms, gravity is completely negligible.
So why does gravity run the universe?
If gravity is so weak, why does it hold planets, stars and galaxies together?
- Mass is always positive, so gravity always adds up. Every kilogram of Earth pulls in the same direction.
- Charge comes in two signs, and matter is almost perfectly neutral. The pulls and pushes of the positive and negative charges in a planet nearly cancel.
- Both forces have unlimited range (1/r²), but only gravity has nothing to cancel it.
So electric forces rule atoms, molecules, friction and every push you feel, while gravity rules anything the size of a moon or bigger.
Worked exampleExample 2: Replacing gravity with electricity
Suppose Earth and the Moon both carried the same charge Q. How large would Q have to be for their electric repulsion to cancel their gravitational attraction? (M_E = 5.97 × 10²⁴ kg, M_M = 7.35 × 10²² kg.)
- Set the forces equal: kQ²/r² = GM_E M_M/r². The r² cancels again.
- Q² = GM_E M_M/k = (6.67 × 10⁻¹¹)(5.97 × 10²⁴)(7.35 × 10²²) / (8.99 × 10⁹).
- Numerator: 2.927 × 10³⁷. Divide by k: 3.256 × 10²⁷ C².
- Q = √(3.256 × 10²⁷) = 5.71 × 10¹³ C.
Q ≈ 5.7 × 10¹³ C each. That is a huge charge, but still a tiny fraction of the charge carried by all the protons in the Moon. A small imbalance would beat gravity, which is why large bodies must be almost perfectly neutral.
Where magnetic fields are different
Magnetic fields share some features with the other two (they act at a distance, they store energy, field lines show direction and strength), but they break the pattern in several ways:
- Source: moving charges (currents, spinning electrons), not charge or mass alone.
- Who feels it: only moving charges. A charge at rest feels no magnetic force.
- Direction of force: perpendicular to both v and B (a hand rule), not along the field lines. That is why it bends paths into circles but never changes speed.
- Shape: field lines form closed loops. No magnetic "monopole" (lone N or S pole) has ever been found.
- Distance: a long straight wire gives B = μ₀I/(2πr), which falls off as 1/r, not 1/r². A solenoid gives a nearly uniform field inside, like parallel plates do for electric fields.
In fact, electric and magnetic fields are two faces of one thing, the electromagnetic field. Whether you see an electric or a magnetic field can depend on how you are moving, a link Einstein explored in his 1905 relativity paper (see Special Relativity). Light itself is a travelling electromagnetic wave (see The Wave Nature of Light).
The four fundamental forces
Every force you have studied (normal force, tension, friction, magnetic attraction) traces back to just four fundamental forces:
- Gravitational: acts on everything with mass or energy; always attractive; unlimited range; by far the weakest.
- Electromagnetic: acts on charges; attracts or repels; unlimited range. Holds atoms and molecules together and causes all contact forces.
- Strong nuclear: holds quarks together inside protons and neutrons, and holds protons and neutrons together in nuclei, overpowering the electric repulsion of the protons. Very short range (about the size of a nucleus).
- Weak nuclear: changes one kind of particle into another, as in beta decay and the nuclear reactions that power the Sun. Even shorter range.
In the Standard Model of particle physics, a force between two particles is carried by exchange particles passing between them: photons for electromagnetism, gluons for the strong force, and W and Z bosons for the weak force. The W and Z were discovered at CERN in 1983. The Standard Model also predicted the Higgs boson, found at CERN in 2012. The Quantum Mechanics unit looks at the Standard Model's particles in more detail.
Two theories that do not quite fit together
The Standard Model leaves gravity out. Our best theory of gravity is Einstein's general relativity (1915), which tells a different story:
- There is no "gravitational force" in the usual sense. Mass and energy curve spacetime, and objects (even light) follow the straightest possible paths through it. Earth orbits the Sun because spacetime near the Sun is curved.
- It explains things Newton's law cannot, such as the slow extra turning of Mercury's orbit and the amount by which starlight bends near the Sun (measured during a 1919 solar eclipse).
- It predicts black holes (see Gravitational Potential Energy, Escape Speed and Black Holes) and gravitational waves. On 14 September 2015, the two LIGO detectors in the United States picked up gravitational waves from two black holes, each roughly 30 times the Sun's mass, merging more than a billion light-years away. Many more mergers have been recorded since by LIGO and partner detectors in Europe and Japan.
- Clocks run at different rates at different heights in a gravitational field, a real effect that GPS satellites have to correct for (next lesson).
The trouble is that general relativity treats gravity as smooth geometry, while the Standard Model treats forces as exchanges of particles (quanta). Physicists have proposed a graviton to carry gravity, but it would be so weakly interacting that no experiment has come close to detecting one. Joining the two theories, often called quantum gravity, is one of the biggest open problems in physics. Newton's law, which you used all unit, remains an excellent approximation whenever gravity is weak and speeds are much less than the speed of light.
Three questions for any field
When you meet a new field problem, ask three questions: What is the source? (mass, charge or moving charge.) Who feels the force? (any mass, any charge, or only moving charges.) Which way does the force point? (along the field, or perpendicular to it.) The answers tell you which equations to reach for.
Just for funEvery time a small magnet lifts a paperclip off the table, it is beating the gravitational pull of the entire Earth, all 5.97 × 10²⁴ kg of it. Gravity is powerful only because there is so much mass on its side.
Hands-on activityField fingerprints at home
Test the three fields with everyday objects and compare them.
You need: a balloon, small bits of tissue paper, a fridge magnet, a few paperclips, a sheet of paper, a piece of cardboard, a piece of aluminium foil and a steel lid or baking tray.
- Electric: rub the balloon on dry hair or a wool sweater and hold it above the tissue bits. Record the largest gap the bits jump across.
- Magnetic: hold the magnet above a paperclip and record the largest gap it jumps across. Then try a coin and a piece of tissue. Which objects respond?
- Shielding: put the paper, then the cardboard, then the foil, then the steel between the magnet and a paperclip. Does each one block the magnetic pull? (Steel should weaken it the most.) Repeat with the charged balloon and the foil: hold the foil near the tissue bits with your hand touching it.
- Gravity: can anything you own block gravity? (Nothing can.)
- Write a short paragraph comparing the three fields: source, which objects respond, whether they attract or repel, and whether they can be blocked.
What to look for: the balloon attracts tissue (and almost any light object) because charges in the tissue shift; the magnet attracts only some metals (iron and steel); the steel tray weakens the magnetic pull because it carries the field lines through itself; the foil connected to your hand reduces the balloon's effect; gravity cannot be shielded.
Grown-up note: keep magnets away from young children (swallowed magnets are dangerous), pacemakers, bank cards and phones.
Your turn
1Name the exchange particle that carries the electromagnetic force, and the one that carries the strong force.
Electromagnetic: the photon. Strong: gluons.
2Give two similarities and two differences between gravitational and electric fields.
Similarities (any two): both follow an inverse-square law for a point source; both have unlimited range; both are conservative and have a potential energy; both have a uniform version (near Earth, between plates). Differences (any two): gravity only attracts, electric forces attract or repel; mass is always positive, charge has two signs; G is far smaller than k; electric fields can be shielded, gravity cannot.
3Two protons are close together. Find the ratio of the electric force to the gravitational force between them. Does it depend on their separation?
Fe/Fg = ke²/(Gm_p²) = 2.301 × 10⁻²⁸ / ((6.67 × 10⁻¹¹)(1.67 × 10⁻²⁷)²) = 2.301 × 10⁻²⁸ / 1.860 × 10⁻⁶⁴ = 1.24 × 10³⁶. Both forces fall off as 1/r², so the ratio does not depend on separation.
4What upward electric field would hold an electron up against gravity near Earth's surface? Which way must the field point?
eE = mg, so E = mg/e = (9.11 × 10⁻³¹)(9.8)/(1.60 × 10⁻¹⁹) = 5.6 × 10⁻¹¹ N/C. The force on an electron is opposite to E, so the field must point downward for the force to point up. It is a tiny field: gravity barely matters for electrons.
5Why do magnetic field lines form closed loops while the electric field lines of a single point charge do not?
Electric field lines start on positive charges and end on negative charges (or go to infinity). There are no magnetic monopoles: every N pole comes with an S pole, so magnetic lines have nowhere to start or end and must close on themselves.
6A nucleus contains several protons squeezed very close together. Why does it not fly apart?
The strong nuclear force between protons and neutrons is much stronger than the electric repulsion at nuclear distances (about 10⁻¹⁵ m). Beyond that tiny range the strong force fades quickly.
More practice: warm-up
1List the four fundamental forces.
Gravitational, electromagnetic, strong nuclear, weak nuclear.
2Which two fields follow an inverse-square law for a point source?
The gravitational field of a point mass and the electric field of a point charge.
3Give the units of gravitational field strength and electric field strength.
g: N/kg (equivalent to m/s²). E: N/C (equivalent to V/m).
4Which fundamental force is always attractive?
5What did the LIGO detectors first observe in September 2015?
Gravitational waves, from two black holes merging.
More practice: on your own
1Explain why gravitational potential energy Eg = −GMm/r is always negative, but electric potential energy Ee = kQq/r can be positive or negative.
Masses are always positive, so GMm > 0 and the minus sign (zero at infinity, attractive force) makes Eg always negative. In kQq/r, the charges can have either sign: like charges give positive Ee (repulsion; work was needed to push them together), unlike charges give negative Ee (attraction, like gravity).
2In a hydrogen atom (r = 5.29 × 10⁻¹¹ m), find the electric force and the gravitational force between the electron and proton.
Fe = ke²/r² = 2.301 × 10⁻²⁸ / (5.29 × 10⁻¹¹)² = 2.301 × 10⁻²⁸ / 2.798 × 10⁻²¹ = 8.22 × 10⁻⁸ N. Fg = Gm_e m_p/r² = 1.015 × 10⁻⁶⁷ / 2.798 × 10⁻²¹ = 3.63 × 10⁻⁴⁷ N. Ratio about 2.3 × 10³⁹, as in Example 1.
3A dust grain of mass 1.0 × 10⁻¹⁵ kg carries 5 extra electrons in Earth's fair-weather electric field of about 100 N/C. Compare the electric and gravitational forces on it.
Fe = qE = 5(1.60 × 10⁻¹⁹)(100) = 8.0 × 10⁻¹⁷ N. Fg = mg = (1.0 × 10⁻¹⁵)(9.8) = 9.8 × 10⁻¹⁵ N. Gravity is about 120 times larger here: with only a few extra electrons on a relatively massive grain, gravity wins. Electric forces win when the charge-to-mass ratio is large (electrons, ions).
4Explain why gravity, the weakest fundamental force, controls the motion of planets and galaxies.
Mass has only one sign, so the gravitational pulls of all the particles in a planet add up. Charge has two signs and large objects are almost exactly neutral, so their electric forces cancel. The strong and weak forces have tiny ranges. Only gravity has both unlimited range and nothing to cancel it.
5You move twice as far from (a) a small planet, (b) a point charge and (c) a long straight current-carrying wire. By what factor does each field change?
(a) g ∝ 1/r²: becomes ¼. (b) E ∝ 1/r²: becomes ¼. (c) B = μ₀I/(2πr) ∝ 1/r: becomes ½. The wire is a line source, not a point source, so its field fades more slowly.
Challenge
1Two 1.0 kg metal spheres are 1.0 m apart. How many extra electrons would each need so that their electric repulsion exactly balances their gravitational attraction?
kQ²/r² = Gm²/r², so Q = m√(G/k) = (1.0)√(6.67 × 10⁻¹¹ / 8.99 × 10⁹) = √(7.42 × 10⁻²¹) = 8.61 × 10⁻¹¹ C. Number of electrons: 8.61 × 10⁻¹¹ / 1.60 × 10⁻¹⁹ = 5.4 × 10⁸. A kilogram of matter contains roughly 10²⁶ electrons, so this is an imbalance of only about 1 part in 10¹⁸.
2Charges of +9.0 μC and −1.0 μC are 0.40 m apart. (a) Find the point on the line through them where E = 0. (b) Explain why two masses have a zero-field point between them, but these two charges do not.
(a) Between the charges, both fields point toward the −1.0 μC charge, so they cannot cancel there. Look beyond the smaller charge, a distance x from it: k(9.0)/(0.40 + x)² = k(1.0)/x², so 3x = 0.40 + x and x = 0.20 m beyond the −1.0 μC charge (0.60 m from the +9.0 μC charge). (b) Between two masses, each field points toward its own mass, so they point opposite ways and can cancel. With opposite charges, between them both fields point the same way (from + toward −). Gravity behaves like a pair of like-sign "charges" that attract instead of repel.
3Newton's law of gravitation and general relativity both predict that the Moon orbits Earth. Describe how each theory explains it, and give one observation that only general relativity explains.
Newton: Earth exerts a force F = GMm/r² on the Moon, toward Earth's centre; this provides the centripetal force for the orbit. General relativity: Earth's mass curves spacetime, and the Moon follows the straightest possible path (a geodesic) through that curved spacetime, which looks like an orbit to us; no force is needed. Observations only GR explains (any one): the extra precession of Mercury's orbit; the full amount of starlight bending near the Sun; gravitational waves; clocks running faster at higher altitude (GPS).
Quiz
1. Which particle carries the electromagnetic force in the Standard Model?
2. Which property belongs to magnetic fields but not to gravitational or electric fields?
3. The ratio of the electric force to the gravitational force between two protons
4. Which force holds protons together inside a nucleus despite their electric repulsion?
5. In general relativity, gravity is described as
6. In the analogy between gravitational and electric fields, the gravitational quantity that corresponds to electric charge is
Big idea (D3.1, D3.2): gravitational and electric fields have the same inverse-square form, with mass playing the role of charge; magnetic fields act only on moving charges and form closed loops. All forces come from four fundamental interactions, explained by two different theories: the Standard Model (exchange particles) and general relativity (curved spacetime).
How to help: Ask your teen to explain the "dictionary" table without looking, then to say why gravity, the weakest force, controls the solar system. The field fingerprints activity is a good way to start the conversation.
Common difficulties:
- Thinking gravity is strong because it holds planets in orbit. It is by far the weakest force; it wins only because mass never cancels.
- Treating magnetic fields as just another inverse-square field.
- Mixing up which particle carries which force.
Safety: small magnets are dangerous if swallowed; keep them away from young children and from medical devices such as pacemakers.