How can Maxwell's equations be correct if the terms have different units on both sides of the equation?

Question

For instance, the divergence of the magnetic field has units but it's equated to zero?

Answer 1

How can Maxwell's equations be correct if the terms have different units on both sides of the equation?

Your premise is completely incorrect. Maxwell's equations always have the same units on both sides of the equation. There's two components two this:

• Three of Maxwell's equations are traditionally phrased as equalities between dimensional quantities:

  • $\nabla \times \mathbf E = -\frac{\partial\mathbf B}{\partial t}$ has units of $\mathrm{V/m^2}$ on the left and $\mathrm{T/s}$ on the right, and both of those units coincide.
  • $\nabla \times \mathbf B = \mu_0\mathbf J + \mu_0\varepsilon_0 \frac{\partial\mathbf E}{\partial t}$ has units of $\mathrm{T/m}$ on the left and $\mathrm{N/Am^2}$ or $\mathrm{V\:s/m^3}$ on the right, and all of those units coincide.
  • $\nabla \cdot \mathbf E = \frac{1}{\varepsilon_0}\rho$ has units of $\mathrm{V/m^2}$ on the left and $\mathrm{C/F\:m^2}$ on the right, and both of those units coincide.

• The fourth Maxwell equation, the magnetic divergence law $\nabla \cdot\mathbf B=0$, is perfectly consistent, because zero is zero regardless of what units it is applied to.* If you will you can see the right-hand-side of that equation as $0\:\mathrm{T/m}$, but really, you gain nothing by that.

  For more on the physical dimensionality of zero, see Is $0\,\mathrm m$ dimensionless? and Should zero be followed by units?, and the many linked questions on the right-hand sidebar from those two.

^{* unless you're talking about pathological cases like degrees Celsius or Fahrenheit, which are obviously excluded here.}