Question

Why does the induction equation of the magnetohydrodynamics assume the electric field E

is time t
independent or ∂E∂t=0
?

Note: I am querying about the general case described in the Section Mathematical statement of the Wikipedia page linked to earlier, not the special case of infinite conductivity describe in the subsequent section Perfectly-conducting limit.

Specifically, Maxwell's equation stipulates
∇×B⃗ =μ0(J⃗ +ε0∂E⃗ ∂t).
The ∂E⃗ ∂t
term would produce a ∂2B⃗ ∂t2
term giving a wave component which is currently missing in the final PDE for B⃗
.

The derivation in the first paragraph has
∇×B⃗ =μ0J⃗ .
Where did the ∂E∂t
term go?

A) The ∂E/∂t term is considered negligible in typical MHD scenarios involving plasma dynamics.
B) Time independence of E simplifies the equations for magnetohydrodynamics without significantly impacting accuracy in many physical situations.
C) In the derivation process, the ∂E/∂t term is incorporated differently to account for its effects in the final partial differential equation (PDE) for B.
D) The assumption of time-independence for E aligns with experimental observations and empirical data gathered in most magnetohydrodynamic studies.

Answer 1

The induction equation of magnetohydrodynamics assumes that the electric field E is time-independent (∂E/∂t=0) because it is considered negligible in typical MHD scenarios involving plasma dynamics. This assumption simplifies the equations for magnetohydrodynamics without significantly impacting accuracy in many physical situations.

Step-by-step explanation:

The induction equation of magnetohydrodynamics assumes that the electric field E is time-independent (∂E/∂t=0) because it is considered negligible in typical MHD scenarios involving plasma dynamics. This assumption simplifies the equations for magnetohydrodynamics without significantly impacting accuracy in many physical situations. The ∂E/∂t term may be incorporated differently in the derivation process to account for its effects in the final partial differential equation (PDE) for the magnetic field B.