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Formula: **3. Maxwell equation (differential form)**

$$\nabla \times \class{purple}{\boldsymbol{E}} ~=~ -\frac{\partial \class{violet}{\boldsymbol{B}}}{\partial t}$$

## Electric field

`$$ \class{purple}{\boldsymbol E} $$`Unit

`$$ \frac{\mathrm{V}}{\mathrm{m}} = \frac{\mathrm{N}}{\mathrm{C}} = \frac{\mathrm{kg} \, \mathrm{m}}{\mathrm{A} \, \mathrm{s}^3} $$`

Electric field indicates how large and in what direction the electric force on a charge would be if that charge were placed at location \((x,y,z)\).

In the third Maxwell equation in differential form, on the left-hand side is the rotation \( \nabla \times \class{blue}{\boldsymbol{E}} \) of the electric field, i.e. the cross product between nabla operator \(\nabla\) and E-field. This rotating field corresponds to the negative time derivative of the magnetic field \( \class{violet}{\boldsymbol{B}} \).

## Magnetic field

`$$ \class{violet}{\boldsymbol{B}} $$`Unit

`$$ \mathrm{T} = \frac{\mathrm{kg}}{\mathrm{A} \, \mathrm{s}^2} $$`

Magnetic flux density determines the force on a moving electric charge.

The minus sign in front of the time derivative of the magnetic field accounts for the Lenz rule. All in all, the third Maxwell equation states that a magnetic field changing in time causes a rotating electric field and vice versa. It is therefore the law of induction in its general form.