Past exam of the mathematics course of the University of Cambridge / 2016 / ia / Paper 4 / 3B (Dynamics and Relativity)

Use Newton's second law, $F=ma$. Acceleration has physical dimension $L{T}^{-2}$, so multiplying by mass gives the dimension of forceThe bracket notation records a physical dimension, rather than the numerical value or the choice of units.

The torque about an origin is $\mathbf{\tau }=\mathbf{r}\times \mathbf{F}$. A cross product multiplies the dimensions of its factors, so torque has the physical dimension of length times force:

The work done by a force is $W=\int \mathbf{F}\cdot d\mathbf{r}$. By the work-energy theorem, this has the same physical dimension as energy. ThusAlthough torque and energy have the same physical dimension, they represent different physical quantities.

From Newton's law of universal gravitation, $F=G{m}_1{m}_2/r^2$, the gravitational constant has physical dimensionHere the two factors of mass and the inverse square of the separation determine the powers of $M$ and $L$.

An electric field is defined through the electric force on a stationary electric charge: $\mathbf{F}=q\mathbf{E}$. Therefore the electric-field dimension is

For the magnetic part of the Lorentz force, $\mathbf{F}=q\mathbf{v}\times \mathbf{B}$. Since velocity has physical dimension $L{T}^{-1}$, the magnetic field has dimension

Use Coulomb's law in vacuum, $F=q_1{q}_2/(4\pi {ϵ}_0{r}^2)$. The constant $4\pi$ is dimensionless, so the vacuum-permittivity dimension isThus the physical dimension of vacuum permittivity follows directly from the dimensions of electric charge, force, and separation.