# Electromagnetism: follow the flux and the moving boundary — Study worksheet

[Study guides](https://unireadinglist.com/guides/electromagnetic-reasoning/)

[Course topic: Electromagnetism](https://unireadinglist.com/subjects/physics/#course-electromagnetism)

Sources checked: 2026-11-04

Study route and exercise: University Reading List editorial project.

A calculus-based route from magnetic flux to Faraday induction. OpenStax Volume 2 supplies a first-course entry, Griffiths fifth edition an advanced undergraduate spine, and Fleisch a focused Maxwell-equation supplement. The guide uses prescribed ideal fields and loops, SI units and stated orientation. It does not design hardware or infer fields from a complete source model.

## Choose your starting point

- [ ] I need a free first course
  Use OpenStax after introductory mechanics and single-variable calculus; its thermodynamics chapters are separate from the electromagnetism route.

- [ ] I am ready for vector-field analysis
  Use Griffiths when line/surface integrals and vector calculus can support a full electrodynamics course.

- [ ] I need to understand each Maxwell term
  Use Fleisch beside a main course, rather than expecting a complete electricity-and-magnetism syllabus.

## Compare the three books

### University Physics Volume 2

William Moebs; Samuel J. Ling; Jeff Sanny · 2016
Free calculus-based entry

Choose it when: Flux, induction and circuit signs are new and you need worked first-course explanations.
Use another route when: You already need a full differential-field boundary-value treatment.
Selected reading: Magnetic flux, Faraday’s law, Lenz’s law and induced electric fields in the induction material; Maxwell equations afterward.
Limits and trade-offs: The full volume also covers thermal physics; selected induction reading is a route through part of it.

#### Try before choosing

For uniform B normal to a fixed area, write flux and the sign of emf before inserting numbers.
Check your work: The area vector fixes the sign. Flux has units Wb, emf Wb/s=V.

Your notes:

________________________________

ISBN: 9781938168161

[Official free reading](https://openstax.org/details/books/university-physics-volume-2)

[Book record and access options](https://unireadinglist.com/subjects/physics/university-physics-volume-2/)

### Introduction to Electrodynamics

David J. Griffiths · 5 · 2024
Advanced undergraduate field spine

Choose it when: You can integrate a nonuniform field and distinguish local equations from integral statements.
Use another route when: Curl, divergence and oriented line/surface integrals are still opaque.
Selected reading: Vector analysis as needed, followed by electrodynamics, induction/emf and Maxwell-equation material in the fifth edition.
Limits and trade-offs: It offers a broader mathematical course but expects more than scalar plug-in formulas; a preview is not the complete text.

#### Try before choosing

Integrate the worked example’s field over x instead of multiplying by its left-edge value.
Check your work: The linear spatial term contributes alpha L²/2; omitting it gives the wrong emf magnitude.

Your notes:

________________________________

ISBN: 9781009397759

[Book record and access options](https://unireadinglist.com/subjects/physics/introduction-to-electrodynamics/)

### A Student's Guide to Maxwell's Equations

Daniel Fleisch · 2008
Focused meaning of Maxwell equations

Choose it when: You can follow basic calculus but want each flux, circulation and time-derivative term explained.
Use another route when: You need a complete circuit, electrostatic boundary-value or radiation textbook.
Selected reading: Faraday’s law, with the electric/magnetic Gauss-law chapters for flux and the wave-equation chapter later.
Limits and trade-offs: Its narrow focus complements a main book; companion audio and problems are not the entire textbook made free.

#### Try before choosing

Explain why a static field can give emf when the conducting boundary moves.
Check your work: Distinguish the total flux derivative for the moving loop from the partial time derivative at a fixed point.

Your notes:

________________________________

ISBN: 9780521701471

[Book record and access options](https://unireadinglist.com/subjects/physics/students-guide-to-maxwells-equations-fleisch/)

## Check your preparation

### Can I orient the circuit?

Your notes:

________________________________

### Can I integrate before differentiating?

Your notes:

________________________________

### Can I separate model assumptions?

Your notes:

________________________________

## A six-step study plan

- [ ] Draw the oriented geometry

Selected reading: Your chosen flux/Gauss-law material.

What to do: Draw x,y,z, the rectangle, its normal and positive circulation.

Check your work: The sign convention is visible before any minus sign is used.

Your notes:

________________________________

- [ ] Check dimensions

Selected reading: First-course magnetic-flux material.

What to do: Build a units table for B, b0, alpha, flux and emf in the worked example.

Check your work: b0 has T/s and alpha has T/(m·s) when the field contains a factor t.

Your notes:

________________________________

- [ ] Integrate a nonuniform field

Selected reading: Griffiths vector analysis or matching integral tools.

What to do: Compute the exact flux and compare it with a left-edge approximation.

Check your work: The approximation omits a nonzero spatial contribution.

Your notes:

________________________________

- [ ] Apply the induction law

Selected reading: Faraday and Lenz material.

What to do: Differentiate the flux, find signed current and check resistive power.

Check your work: Magnitude and clockwise/counterclockwise direction agree with increasing +z flux.

Your notes:

________________________________

- [ ] Move the boundary

Selected reading: Induction/emf material and the follow-up.

What to do: Compute the derivative of flux for L(t), and verify it using the moving-side motional term.

Check your work: A zero partial time derivative of B does not force zero emf.

Your notes:

________________________________

- [ ] Audit the argument

Selected reading: Your two complete solutions.

What to do: Write which derivative and which contour each formula refers to.

Check your work: Explain why the fixed-loop and moving-loop calculations use different field and geometry assumptions.

Your notes:

________________________________

## Worked exercise: A nonuniform field cannot be replaced by its edge value

A fixed rectangular loop spans 0≤x≤L=0.50 m and 0≤y≤w=0.20 m, with normal +z. The prescribed field is B\_z(x,t)=\[b0+alpha x\]t, b0=0.20 T/s, alpha=0.40 T/(m·s). Its resistance is 3.0 Ω. Find flux, signed emf, current and resistive power.

Assumptions: The field is normal to the loop and independent of y; SI units. Positive circulation is counterclockwise viewed from +z. A quasistatic ideal resistive loop is used, with self-inductance and its influence on the prescribed external field neglected.

Your notes:

________________________________

## Change an assumption: A static field with a moving side still induces emf

Replace the field by the static B\_z(x)=0.20 T+(0.40 T/m)x. Keep width w=0.20 m and the left side at x=0, but move the right side so L(t)=0.50 m+(0.10 m/s)t. Find the signed emf at t=0. Do not carry over the original fixed-loop geometry or its field units.

Your notes:

________________________________

## Before moving on

- Compute an oriented flux integral with units.
- Explain the difference between field change and boundary motion.
- Check the induced direction and power without changing sign conventions.

## Answer and interpretation

### Can I orient the circuit?

Choose +z as surface normal. Positive boundary circulation is counterclockwise viewed from +z; a negative emf is clockwise in that view.

### Can I integrate before differentiating?

For B\_z(x,t) independent of y, flux through 0≤x≤L, 0≤y≤w is w∫B\_z dx. A point sample works only under additional conditions.

### Can I separate model assumptions?

State whether the loop is fixed, whether B is prescribed and whether self-inductance is neglected. Resistance converts emf into current only in the specified resistive model.

### A nonuniform field cannot be replaced by its edge value

1. Integrate: Phi(t)=w t\[b0 L+alpha L²/2\]. The bracket is 0.20·0.50+0.40·0.50²/2=0.15 T·m/s, so Phi(t)=(0.030 Wb/s)t.
2. Faraday’s law gives emf=−dPhi/dt=−0.030 V. The negative sign means clockwise circulation viewed from +z, opposing the increasing +z flux.
3. With the stated resistive model, I=emf/R=−0.010 A. Dissipation is I²R=0.00030 W=0.30 mW, always nonnegative.

Phi(t)=(0.030 Wb/s)t; emf=−0.030 V; I=−0.010 A; power=0.30 mW.

Test the result: Using only B at x=0 gives −0.020 V and misses the gradient contribution. For this linear field only, the midpoint value times area reproduces the integral exactly; that shortcut is not general.

### Change an assumption: A static field with a moving side still induces emf

1. The flux is Phi(t)=w\[(0.20 T)L(t)+(0.40 T/m)L(t)²/2\]. The field has no explicit time dependence, but the integration boundary does.
2. Differentiate: dPhi/dt=w\[0.20 T+(0.40 T/m)L(t)\]L′(t). At t=0 this is 0.20·0.40·0.10=0.0080 Wb/s, giving emf=−0.0080 V.
3. On the moving right side, v is +x and B is +z, so v×B points −y. Positive contour direction there is +y; the motional contribution is −v B(L)w=−0.0080 V.

The signed emf at t=0 is −8.0 mV, clockwise viewed from +z, despite partial B/partial t=0.

Test the result: Setting the side speed to zero makes this emf vanish. Only the right side moves in the stated geometry; do not add two moving-side contributions or substitute the original T/s coefficients.

## Common mistakes

### Replacing flux by a field sample

Nonuniform fields require an integral, unless a justified averaging rule applies.

### Using an unsigned minus sign

A direction statement needs a surface normal and corresponding contour orientation.

### Confusing partial and total change

A static field does not mean the flux through a moving loop is constant.

## Sources and edition checks

- [OpenStax: calculus-based volume2 preface](https://openstax.org/books/university-physics-volume-2/pages/preface)
  The official volume provides a calculus-based entry and distinguishes original publication from current digital notices.

- [Cambridge: fifth-edition front matter](https://assets.cambridge.org/97810093/97759/frontmatter/9781009397759_frontmatter.pdf)
  Cambridge identifies fifth edition, selected hardcover and the vector-analysis/electrodynamics reading scope.

- [Cambridge: Student Guide series catalogue](https://assets.cambridge.org/isbn13/97805219/49620/full_version/9780521949620_pub.pdf)
  Cambridge’s catalogue identifies the 2008 paperback and focused chapters on each Maxwell law and the wave equation.

These are proposed study activities, not official course assignments or a claim that every book has been read cover to cover.

[How we curate](https://unireadinglist.com/editorial/)