18.3 Electric force between point charges

1. Modelling Spherical Conductors

In electrostatics, we rarely deal with infinitesimally small “points”, we deal with charged spheres.

The point charge assumption

• For any point outside a spherical conductor, the charge on the sphere is treated as if it were a single point charge at the sphere’s centre.
• The electric field lines around a spherical conductor are therefore identical to those around a point charge.

Electric field lines around a uniform spherical conductor are identical to those on a point charge

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Coulomb’s Law

Statement of Coulomb's Law

The electric force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of their separation.

The Equation

From Coulomb’s Law:

$$\begin{array}{rl}\mathrm{F}& \propto \frac{Q_1{Q}_2}{r^2}\\ \mathrm{F}& =k\frac{Q_1{Q}_2}{r^2}\end{array}$$

• Where $k$ is the constant of proportionality, $k=8.99\times 10^9\mathrm{m}{\mathrm{F}}^{-1}$, provided that the charges are in a vacuum.
• However, if the charges are in air rather than a vacuum, the change to the value of $k$ is negligible, therefore $k=8.99\times 10^9\mathrm{m}{\mathrm{F}}^{-1}$ can still be used.

The value $k$ can be expressed in terms of another constant:

$$k=\frac{1}{4\pi {ϵ}_{\circ }}$$

• Where ${ϵ}_{\circ }$ is the permittivity of free space, ${ϵ}_{\circ }=8.85\times 10^{-12}\mathrm{F}{\mathrm{m}}^{-1}$.
• The constant ${ϵ}_{\circ }$ relates the electric field tot he magnetic field and the propagation of electromagnetic waves.

Newton’s Third Law Application

The force exerts a symmetrical effect,

$$|{\mathrm{F}}_{Q_{1}on{Q}_2}|=|{\mathrm{F}}_{Q_{2}on{Q}_1}|$$

Even if $Q_1=+1\mathrm{C}$ and $Q_2=+1000\mathrm{C}$, the force felt by the small charge is exactly the same magnitude as the force felt by the large charge.

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Comparison: Electric vs Gravitational Fields

Feature	Electric Force	Gravitational Force
Formula	$F=\frac{Q_1{Q}_2}{4\pi {ϵ}_0{r}^2}$	$F=\frac{G{M}_1{M}_2}{r^2}$
Relation to Distance	Inverse Square Law ($F\propto r^{-2}$)	Inverse Square Law ($F\propto r^{-2}$)
Range	Infinite	Infinite
Nature of Force	Attractive or Repulsive	Always Attractive
Magnitude	Very Strong ($k\approx 10^9$)	Very Weak ($G\approx 10^{-11}$)
Constant	Depends on the medium ($ϵ$)	Universal ($G$ is constant everywhere)