Suppose you are trying to fix Ampere's Law, so that ∇⃗· (∇⃗×B⃗) = 0, as it must. You hypothesize

Question

Suppose you are trying to fix Ampere's Law, so that $\vec{\nabla} \cdot (\vec{\nabla} \times \vec{B}) = 0$, as it must. You hypothesize that the form: $\vec{\nabla} \times \vec{B} = \mu_0 \vec{J} + \vec{X}$ will work (at least for a vacuum). Show that the choice of $\vec{X} = \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}$ does the job.

          Suppose you are trying to fix Ampere's Law, so that $\vec{\nabla} \cdot (\vec{\nabla} \times \vec{B}) = 0$, as it must. You hypothesize that the form:
$\vec{\nabla} \times \vec{B} = \mu_0 \vec{J} + \vec{X}$
will work (at least for a vacuum). Show that the choice of $\vec{X} = \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}$ does the job.

Suppose you are trying to fix Ampere's Law, so that ∇⃗· (∇⃗×B⃗) = 0, as it must. You hypothesize that the form:
∇⃗×B⃗ = μ0 J⃗ + X⃗
will work (at least for a vacuum). Show that the choice of X⃗ = μ0 ϵ0 (∂E⃗)/(∂ t) does the job.

Added by Jorge C.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Madhur L

06/21/2023

Step 1

Step 1: Start with the given equation ∇ × B = μ0J + X. Show more…

Show all steps

Thanks for your feedback!

Suppose you are trying to fix Ampere's Law, so that ∇ × B = 0, as it must. You hypothesize that the form: ∇ × B = μ0J + X will work (at least for a vacuum). Show that the choice of X = μ0εE/∂t does the job.