Download Notes–Maxwell`s Equations

Equation
𝒒𝑒𝑛𝑐𝑙
∮ 𝑬 ∙ 𝑑𝑨 =
𝜀𝑜
∮ 𝑩 ∙ 𝑑𝑨 = 0
∫ 𝑬 ∙ 𝑑𝑟 = −
𝑑Φ𝐵
𝑑𝑡
Maxwell’s Equations of Electromagnetism
Name
Meaning
Gauss’ Law for Electricity
Relates net electric flux
Φ𝐸 = ∮ 𝑬 ∙ 𝑑𝑨, to enclosed
charge
Gauss’ Law for Magnetism
Faraday’s Law
(with Lenz’ negative sign)
also
ℇ=−
𝑑Φ𝐵
𝑑𝑡
Relates net electric flux
Φ𝐵 = ∮ 𝑩 ∙ 𝑑𝑨, to enclosed
magnetic system. Mathematically
denies the existence of a
magnetic monopole.
Relates induced electric field
(voltage, induced current, etc.) to
changing magnetic flux. The
negative sign indicating direction
of Emf opposes the change in Bflux.
Notice also the definition of
electric potential (V) on the left
hand side of the top equation.
∮ 𝑩 ∙ 𝑑𝑙 = 𝜇𝑜 𝜀𝑜
𝑑Φ𝐸
+ 𝜇𝑜 𝑖𝑒𝑛𝑐𝑙
𝑑𝑡
∫ 𝑩 ∙ 𝑑𝑙 = 𝜇𝑜 𝑖𝑒𝑛𝑐𝑙
Ampere-Maxwell Law
Ampere’s Law
Relates induced B-field to
changing electric flux and current.
This is the full version of the
equation, we only use the
Ampere’s Law part
When to Use
Normally used to determine the
electric field due to some
geometry of charge. If a
“Gaussian Surface” is picked
carefully such that the E-field has
uniform intensity at all points, E
comes out of the integral. Implies
that if qencl = 0, the E-field must
also be zero.
Never, unless to prove no Bmonopoles.
Not to be confused with a simple
calculation of B-flux for use in
Faraday’s Law. Φ𝐵 = ∫ 𝑩 ∙ 𝑑𝑨
When finding the induced Emf
(voltage), induced current that
results (Emf/Resistance), or the Efield inside the conductor/wire
(usually V=Ed). To mathematically
utilize Faraday’s Law you probably
need to fund a function of the Bflux: Φ𝐵 = ∫ 𝑩 ∙ 𝑑𝑨. The
solutions to this flux integral are
usually:
 B*A(t)
 B(t)*A
 B(t)*A(t)
Ampere-Maxwell: Never for this
class.
Ampere’s Law: To find the B-field
due to a current. Applies with
Biot-Savart equation.