Flux of the Field

Why flux matters here

• The concept of the flux of the magnetic field is vital to understanding Faraday's law: the whole of electromagnetic induction is written in terms of $\phi$ and its rate of change.

Direction assigned to an area element

• Consider a small element of area $\vec{da}$ bounded by a closed curve. A direction is assigned to the element so that if the bounding curve is traversed in the direction of the arrow on it, the normal comes out of the plane of the paper towards the reader.
• Equivalently, it is the direction in which a right-handed screw would advance if rotated in the sense of the arrow on the curve.
• Diagram in words: a small patch of surface with an anticlockwise arrow running round its boundary, and a straight arrow $\vec{da}$ sticking out of the patch towards the reader.

Flux through a small element and through a finite surface

• If the element $\vec{da}$ sits in a magnetic field $\vec{B}$, the scalar quantity

$$d\phi = \vec{B}\cdot\vec{da} = \lvert\vec{B}\rvert\,\lvert\vec{da}\rvert\cos\theta$$

• is called the flux of $\vec{B}$ through the area $\vec{da}$, where $\theta$ is the angle between the direction of $\vec{B}$ and the direction assigned to the area.
• For a finite surface $S$ the field need not be the same at all points, so the surface is divided into small sections $\vec{da}$, the flux through each is calculated, and the results are integrated over the whole area:

$$\phi = \int_S \vec{B}\cdot\vec{da}$$

• $\vec{B}$ cannot be taken outside the integral unless it is the same everywhere on $S$. When it is uniform and the surface is flat, $\phi = BA\cos\theta$.
• Diagram in words: a curved finite surface $S$ bounded by a contour C, ruled into small patches, with an arrow $\vec{da}$ on one patch and field lines of $\vec{B}$ crossing it obliquely.

Time-varying flux and Faraday's law

• If the magnetic field at every point also changes with time, then the flux changes with time:

$$\phi(t) = \int_S \vec{B}(t)\cdot\vec{da}$$

• Faraday's discovery was that the rate of change of flux $\dfrac{d\phi}{dt}$ is related to the work done in taking a unit positive charge once around the contour C in the reverse direction — and that work is just the emf. Accordingly,

$$e = -\dfrac{d\phi}{dt} = -\dfrac{d}{dt}\int_S \vec{B}(t)\cdot\vec{da}$$

• In SI units the emf $e$ is in volt, the flux $\phi$ in weber and the time $t$ in second.
• An important consequence: even if $\vec{B}$ does not change with time, the flux may still vary if the area $S$ changes with time. That is the origin of motional emf in 12.5.

The induced electric field

• Writing the emf as the work done per unit charge round the contour gives the circuital form

$$\oint \vec{E}\cdot\vec{dl} = -\dfrac{d\phi_m}{dt}$$

• where $\vec{E}$ is the electric field induced when the magnetic flux changes at the rate $\dfrac{d\phi_m}{dt}$.
• For a uniform field $B(t)$ pointing upward and filling a circular region of radius $s$ in a horizontal plane, symmetry makes $\vec{E}$ tangential and of constant magnitude on the circle of radius $s$, so $E(2\pi s) = -\pi s^2\dfrac{dB}{dt}$ and

$$E = -\dfrac{s}{2}\dfrac{dB}{dt}$$

Standard results

• $d\phi = \vec{B}\cdot\vec{da} = B\,da\cos\theta$; $\phi = \int_S\vec{B}\cdot\vec{da}$; for a uniform field on a flat area, $\phi = BA\cos\theta$.
• $\phi$ is maximum ($=BA$) when the plane of the coil is perpendicular to $\vec{B}$, and zero when the plane of the coil contains $\vec{B}$.
• $e = -\dfrac{d\phi}{dt}$; the flux can change through $B$, through $A$, or through the orientation $\theta$.
• Induced electric field in a circular region of radius $s$: $E = -\dfrac{s}{2}\dfrac{dB}{dt}$.