31.5.3 The Ampère-Maxwell Law

We have only one equation to go, but this one will now be easier. The Ampère-Maxwell law is concerned with the changing electric flux through a closed curve. FIGURE 31.22 shows a very narrow rectangle in the xz-plane. The electric field is perpendicular to this rectangle; hence the electric flux through it is $Φ_{e} = E_{y} A_{rectangle} = E_{y} l Δ x$ . This flux is changing at the rate

\frac{d Φ_{e}}{d t} = \frac{d}{d t} (E_{y} l Δ x) = \frac{\partial E_{y}}{\partial t} l Δ x

(31.28)

The figure illustrates a narrow rectangle kept in the x z plane.

FIGURE 31.22 Full Alternative Text

The line integral of $\vec{B} \cdot d \vec{s}$ around this closed rectangle is calculated just like the line integral of $\vec{E} \cdot d \vec{s}$ in Figure 31.21. $\vec{B}$ is perpendicular to $d \vec{s}$ on the narrow ends, so $\vec{B} \cdot d \vec{s} = 0$ . The field at all points on the left edge is $\vec{B} (x)$ , and this field is parallel to $d \vec{s}$ to make $\vec{B} \cdot d \vec{s} = B_{z} (x) d s$ . Similarly, $\vec{B} \cdot d \vec{s} = - B_{z} (x + Δ x) d s$ at all points on the right edge, where $\vec{B}$ is opposite to $d \vec{s}$ . Thus, if we let $Δ x \to 0$ ,

\oint^{​} \vec{B} \cdot d \vec{s} = B_{z} (x) l - B_{z} (x + Δ x) l = - ​ ​ [B_{z} (x + Δ x) - ​ ​ B_{z} (x) ​ ​ ​ ​] ​ ​ ​ l = - \frac{\partial B_{z}}{\partial x} l Δ x

(31.29)

Equations 31.28 and 31.29 can now be used in the Ampère-Maxwell law:

\oint \vec{B} \cdot d \vec{s} = - \frac{\partial B_{z}}{\partial x} l Δ x = ϵ_{0} μ_{0} \frac{d Φ_{e}}{d t} = ϵ_{0} μ_{0} \frac{\partial E_{y}}{\partial t} l Δ x

The area of the rectangle cancels, and we’re left with

\frac{\partial B_{z}}{\partial x} = - ϵ_{0} μ_{0} \frac{\partial E_{y}}{\partial t}

(31.30)

Equation 31.30 is a second required condition that the fields must satisfy.