newton

SI coherent derived unit with special name and symbol

Name	Symbol	Quantity	Base units
newton	N	force	kg m s^-2
The newton, symbol N, is the SI coherent derived unit of force. It is the special name for the kilogram metre per second squared, symbol kg m s^-2. One newton is the force needed to accelerate a mass of one kilogram at a rate of one metre per second squared in the direction of the applied force.
Definition			h c^-1 Δν_Cs²
$1 \mspace{4mu} \text{N} \mspace{6mu} = \dfrac{299 \mspace{4mu} 792 \mspace{4mu} 458}{(6.626 \mspace{4mu} 070 \mspace{4mu} 15 \times 10^{-34})(9 \mspace{4mu} 192 \mspace{4mu} 631 \mspace{4mu} 770)^2} \mspace{6mu} \dfrac{h \mspace{4mu} {\Delta \nu _{Cs}}^2}{c}\\ \\ \\ 1 \mspace{4mu} \text{N} \mspace{6mu} \approx 5.354 \mspace{4mu} 081 \mspace{4mu} 104 \mspace{4mu} 982 \mspace{4mu} 697 \mspace{4mu} 161 \times10^{21} \mspace{6mu} h \mspace{4mu} c^{-1} \mspace{4mu} {\Delta \nu _{Cs}}^2$

The newton is named after the English physicist Sir Isaac Newton (1643 – 1727).

Force

A force is any interaction that, when unopposed, will change the motion of an object. A net force applied to an object with mass will cause the object to change its velocity, or accelerate. A force has both magnitude and direction, making it a vector quantity.

Newton’s second law of motion

Newton’s second law of motion states that the rate of change of momentum of a body is directly proportional to the force applied, and this change in momentum takes place in the direction of the applied force.

The law can also be stated in terms of an object’s acceleration. For a constant-mass system, the net force, F, applied to an object of mass, m, is directly proportional to the product of the object’s mass, m, and its acceleration, a.

$F \propto m \mspace{2mu} a$

Using SI coherent units, the proportionality constant is 1. Thus:

$F = m \mspace{2mu} a$

where:

F is the net force applied in newtons, symbol N,
m is the mass of the object in kilograms, symbol kg,
a is the object’s acceleration in metres per second squared, symbol m s^-2.

$1 \ \text{N} = 1 \ \text{kg} \ \text{m} \ \text{s}^{-2}$

Newton’s law of universal gravitation

Newton’s law of universal gravitation states that every point mass attracts every other point mass by a force acting along the line intersecting the two points. The force is directly proportional to the product of the two masses, and inversely proportional to the square of the distance between them.

$F \propto \dfrac{m_1 \mspace{2mu} m_2} {r^2}$

Using SI coherent units, the proportionality constant is the gravitational constant, Thus:

$F = G \mspace{2mu} \dfrac{m_1 \mspace{2mu} m_2} {r^2}$

where:

F is the gravitational force acting between the two objects in newtons, symbol N,
G is the gravitational constant, equal to 6.674 30(15) × 10^-11 N m² kg^-2,
m₁ and m₂ are the masses of the two objects in kilograms, symbol kg,
r is the distance between the two objects in metres, symbol m.

Coulomb’s law

Coulomb’s law states that there is a force of attraction or repulsion acting along a straight line between two point charges. The force is directly proportional to the product of the two charges, and inversely proportional to the square of the distance between them. For like charges, the force is one of repulsion, for opposite charges the force is one of attraction.

$F \propto \dfrac{q_1 \mspace{2mu} q_2}{r^2}$

Using SI coherent units, the proportionality constant is the Coulomb constant, Thus:

$F = k_e \mspace{2mu} \dfrac{q_1 \mspace{2mu} q_2} {r^2}$

where:

F is the electrostatic force acting between the two charges in newtons, symbol N,
k_e is the Coulomb constant, equal to 8.987 551 786 130(14) × 10⁹ N m² C^-2,
q₁ and q₂ are the magnitudes of the two charges in coulombs, symbol C,
r is the distance between the two charges in metres, symbol m.

The Coulomb constant, k_e, can be defined in terms of the electric constant, ε₀, or the magnetic constant, μ₀:

$k_e = \dfrac{1}{4 \mspace{2mu} \pi \mspace{2mu} {\varepsilon}_0} = \dfrac{{\mu}_0 \mspace{2mu} c^2}{4 \mspace{2mu} \pi \mspace{2mu}}$

where:

ε₀ is the electric constant in farads per metre, symbol F m^-1,
μ₀ is the magnetic constant in henries per metre, symbol H m^-1,
c is the speed of light in vacuum in metres per second, symbol m s^-1.

Prior to 2019, the ampere was defined such that the value of the magnetic constant, μ₀, was exactly equal to 4π × 10⁻⁷ H m^-1. This implied an exact value for the Coulomb constant, k_e:

$k_e = c^2 \times 10^{-7} \mspace{4mu} \text{H} \mspace{4mu} \text{m}^{-1}$

This expression can still be used to calculate an approximate value for the Coulomb constant, k_e:

$k_e \approx (299 \mspace{4mu} 792 \mspace{4mu} 458 \mspace{4mu} \text{m} \mspace{4mu} \text{s}^{-1})^2 \times 10^{-7} \mspace{4mu} \text{H} \mspace{4mu} \text{m}^{-1}\ \\ \\ k_e \approx 8.987 \mspace{4mu} 551 \mspace{4mu} 787 \mspace{4mu} 37 \times 10^9 \mspace{4mu} \text{N} \mspace{4mu} \text{m}^2 \mspace{4mu} \text{C}^{-2}$

As a consequence of the 2019 redefinition of the ampere, the electric constant, ε₀, and the magnetic constant, μ₀, are no longer defined with exact numeric values. Their definitive values now both depend on the measured value of the fine-structure constant, α, which is a dimensionless fundamental physical constant that quantifies the strength of the electromagnetic interaction between elementary charged particles.

Using SI coherent units,

${\varepsilon}_0 = \dfrac{1}{2 \mspace{2mu} \alpha} \mspace{2mu} \dfrac{e^2}{h \mspace{2mu} c}$

where:

ε₀ is the electric constant in farads per metre, symbol F m^-1,
h is the Planck constant in joule seconds, symbol J s,
c is the speed of light in vacuum in metres per second, symbol m s^-1,
e is the elementary charge in coulombs, symbol C.

It follows that the Coulomb constant, k_e, can be defined in terms of the fine-structure constant, α:

$k_e \ = \ \dfrac{\alpha}{2 \mspace{2mu} \pi} \dfrac{h \mspace{2mu} c}{e^2} \ = \ \alpha \mspace{2mu} \dfrac{\hbar \mspace{2mu} c}{e^2}$

where:

ħ is the reduced Planck constant in joule seconds, symbol J s.

In the SI, h, c and e are all defined with exact numeric values, which means that the value of the Coulomb constant, k_e, depends on only one measured value – the fine-structure constant, α. The measured numerical value of α is 0.007 297 352 564 3(11).

Thus the value of the Coulomb constant, k_e, can be calculated as follows:

$k_e = \dfrac{(0.007 \mspace{4mu} 297 \mspace{4mu} 352 \mspace{4mu} 564 \mspace{4mu} 3(11))(6.626 \mspace{4mu} 070 \mspace{4mu} 15 \times 10^{-34} \mspace{4mu} \text{J} \mspace{4mu} \text{s})(299 \mspace{4mu} 792 \mspace{4mu} 458 \mspace{4mu} \text{m} \mspace{4mu} \text{s}^{-1})}{(2 \mspace{2mu} \pi)(1.602 \mspace{4mu} 176 \mspace{4mu} 634 \times 10^{-19} \mspace{4mu} \text{C})^2}\\ \\ \\ k_e = 8.987 \mspace{4mu} 551 \mspace{4mu} 786 \mspace{4mu} 130(14) \times 10^9 \mspace{4mu} \text{N} \mspace{4mu} \text{m}^2 \mspace{4mu} \text{C}^{-2}$