AcademyApplying Force Models
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Four Fundamental Interactions

Level 1 - Physics topic page in Applying Force Models.

Principle

Universal gravitation, electromagnetism, and the nuclear interactions are the fundamental interaction models behind macroscopic force laws.

Notation

\(G\)
gravitational constant
\(\mathrm{N\,m^{2}\,kg^{-2}}\)
\(k\)
Coulomb constant
\(\mathrm{N\,m^{2}\,C^{-2}}\)
\(m,m_1,m_2\)
mass or source masses
\(\mathrm{kg}\)
\(q,q_1,q_2\)
charge or source charges
\(\mathrm{C}\)
\(\vec F_g\)
gravitational force
\(\mathrm{N}\)
\(\vec F_E\)
electric force
\(\mathrm{N}\)
\(\vec g\)
gravitational field
\(\mathrm{N\,kg^{-1}}\)
\(\vec E\)
electric field
\(\mathrm{N\,C^{-1}}\)
\(r\)
source separation
\(\mathrm{m}\)

Method

Derivation 1: Identify what property each interaction acts on

A force model starts by identifying the property that couples to the interaction. The same object can participate in several interactions at once.

Gravity couples to mass
\[m\rightarrow\vec F_g\]
Electromagnetism couples to charge
\[q\rightarrow\vec F_E\]
Nuclear interactions are short-range
\[r_{\mathrm{nuclear}}\sim10^{-15}\,\mathrm{m}\]

Derivation 2: Build the inverse-square source law

For point sources or spherically symmetric sources, the effect spreads across spherical area. That geometric spreading is why the source-field strength falls like \(1/r^2\).

Spherical area
\[A=4\pi r^2\]
Universal gravitation
\[F_g=G\frac{m_1m_2}{r^2}\]
Ordinary masses attract each other.
Electric inverse square
\[F_E=k\frac{|q_1q_2|}{r^2}\]

The graph below shows only the distance scaling. Doubling separation leaves one quarter of the force magnitude for any point-source inverse-square model.

1234500.250.50.751rrelative F1/r2
Relative magnitude falls rapidly as separation increases.

Derivation 3: Replace the source by a local field

Fields separate the source from the test object. Once the field is known at a position, the force follows from the property of the object placed there.

Gravity in field form
\[\vec F_g=m\vec g\]
Near a planet, \(\vec g\) is often the most practical gravity model.
Electric field form
\[\vec F_E=q\vec E\]
Charge sign matters
\[q<0\Rightarrow\vec F_E\text{ is opposite }\vec E\]

Rules

These are the compact results from the method above.

Universal gravitation
\[F_g=G\frac{m_1m_2}{r^2}\]
Electric force
\[F_E=k\frac{|q_1q_2|}{r^2}\]
Gravity in field
\[\vec F_g=m\vec g\]
Electric in field
\[\vec F_E=q\vec E\]
Nuclear range
\[r_{\mathrm{nuclear}}\sim10^{-15}\,\mathrm{m}\]

Examples

Question
Two masses interact only through gravity. One mass is tripled while the separation doubles. By what factor does the gravitational force change?
Answer
\[F_g\propto\frac{m_1m_2}{r^2}\]
\[\frac{F'_g}{F_g}=\frac{3}{2^2}=\frac{3}{4}\]
The force becomes three quarters of its original value.

Checks

  • Universal gravitation is attractive for ordinary masses.
  • Electric forces attract for opposite charges and repel for like charges.
  • Inverse-square laws need point-source or spherically symmetric models.
  • Contact forces are electromagnetic in origin but modeled macroscopically as normals, tensions, and friction.

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