## Quasi-static situations

### Maxwell's equations (SI units)

**∇∙E** = ρ/ε_{0}, **∇**×**E** = -∂**B**/∂t,

**∇∙B** = 0, **∇**×**B** = μ_{0}**j** + (1/c^{2})∂**E**/∂t.

### Faraday's law

**∇**×**E** = -∂**B**/∂t**,**

∮_{Γ}** E**∙d**r** = -∂**/**∂t∫_{A}**B**∙**n**
dA.

Define the **flux**

F = ∫_{A}**B**∙**n** dA

and the **electromotive force**

ε = ∮_{Γ}** E**∙d**r**.

Then ε = -∂**F/**∂t**.**

The electromotive force** **is the work done per unit charge (W/q = ε) if
it is moved once around the loop Γ.

Any induced emf tries to oppose the flux changes that produce it. This is **
Lenz's rule**.

In the above integral formulas the "loop" Γ can be **any fixed curve** in
space, i.e. a loop that does not change its shape.

### Motional emf

Consider a **well-defined filamentary circuit** which can change its
shape. For such circuit we may write

ε = -d**/**dt∫_{A}**B**∙**n** dA,

i.e. we can combine the emf due to flux changes and the emf due to shape changes
into one equation. (The partial derivative changes to a total derivative.)

### Quasi-static situations

Quasi-static situations refer to non-static situations in which
electromagnetic radiation can be neglected.

Consider N filamentary circuits. Then the flux through the ith circuit is F_{i}
= ∑_{j=1}^{N }F_{ij},

where F_{ij} = M_{ij}I_{j} (SI units), F_{ij}
= cM_{ij}I_{j} (Gaussian units).

M_{ij} = M_{ji} is the **coefficient of mutual induction**
and

M_{ii} = L_{i} is the **coefficient of self inductance**. We
have

ε_{i} = ∑_{j}M_{ij}∂I_{j}**/**∂t.

For a single filamentary circuit we have ε = -L∂I**/**∂t.

To change the current in a circuit we need an external emf, V_{ext}, to
overcome the induced emf ε.

V_{ext }= L∂I**/**∂t.

The **energy stored in the circuit** is U = ½LI^{2}. For a system of
N circuits we have

U = ½∑_{m=1}^{N} F_{m}I_{m}, or_{
}U = ½∫_{all_space}** j**∙**A** dV, or

U = (1/(2μ_{0}))∫_{all_space}** B**∙**B** dV.