 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ib/paper-3/15d/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

In a region with no current, the Ampère-Maxwell equation and Faraday's law are

\nabla \times B = μ_{0} ϵ_{0} \frac{\partial E}{\partial t}, \nabla \times E = - \frac{\partial B}{\partial t} .

Therefore

\frac{\partial w}{\partial t} = ϵ_{0} E \cdot \partial_{t} E + \frac{1}{μ _{0}} B \cdot \partial_{t} B = \frac{1}{μ _{0}} [E \cdot (\nabla \times B) - B \cdot (\nabla \times E)] = - \frac{1}{μ _{0}} \nabla \cdot (E \times B),

\frac{\partial w}{\partial t} + \nabla \cdot S = 0, S = \frac{1}{μ _{0}} E \times B .

S

 New to topics? Read the docs here!