Integrate an oriented magnetic flux before differentiating

Choose the surface orientation

Magnetic flux is a signed surface account, not the amount of magnetic field at one point. On an open planar surface choose one normal; the positive boundary direction follows the right-hand rule. On a closed surface the normal is outward. For a uniform field and planar area, Phi = B A cos(theta), where theta is measured from the normal, not from the plane. A 0.4 T field through 2 m² at 60 degrees to the normal gives +0.4 Wb; reversing the normal gives −0.4 Wb.

Integrate a spatially varying field

The original rectangle spans x from 0 to L and y from 0 to w = 1 m. Assign Bz(x,t) = B0(1 + alpha x/L)(1 + t/t0), with B0 = 0.2 T and t0 = 2 s. Integrating B dot dA first gives Phi(t) = B0 w L(1 + alpha/2)(1 + t/t0). Using only the field at the left edge discards the gradient; using the midpoint happens to be exact here only because the spatial profile is linear.

Differentiate the correct inventory

For one fixed loop the oriented emf is −dPhi/dt = −B0 w L(1 + alpha/2)/t0. A positive normal field that increases produces a negative circulation: clockwise viewed from +z. If a passive resistive loop were present, its induced field would oppose the increase, not necessarily oppose the existing field. Reversing the chosen normal also reverses the positive circulation convention, so it does not reverse the physical effect.

Distinguish an open surface from a closed one

A nonzero flux through this open rectangle is compatible with zero magnetic flux through a closed surface. For a closed box in the assigned z-directed field, top and bottom contributions cancel because Bz does not depend on z; side contributions vanish. The open-loop emf depends on the changing linked flux, not on inventing a magnetic monopole inside a closed box.

Worked original case

L = 2 m and alpha = 1 give Phi(0) = 0.6 Wb and Phi(2 s) = 1.2 Wb. The derivative is 0.3 Wb/s, so oriented emf is −0.3 V. The left-edge shortcut would predict only 0.4 Wb initially and underestimates the flux by one third.

Investigation

Predict normal-reversal signs before running the model. Derive the area integral, check a zero-gradient limit and compare left-edge versus midpoint sampling. Retain a six-face closed-box account that explains why nonzero open flux does not violate magnetic Gauss law.

Changed-case practice

Sources and boundaries

Original teaching checked 31 October 2026 against current official Unit 13 and the relevant OpenStax discussion. No source exercises or diagrams are reproduced. All records and apparatus are virtual, supplied or generated; no real wiring, switching, magnetic braking or electrical measurement is requested. Public feedback and retained work do not establish independent mastery, supervised practical participation or AP authorization.