Electromagnetism: follow the flux and the moving boundary
A field value at one point is not magnetic flux. Define an oriented surface, integrate the field across it and distinguish a changing field from a changing loop. Those choices determine the sign and the size of an induced emf.
Course topic: Electromagnetism
Study route and exercise: University Reading List editorial project. How we curate · Sources checked:
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Scope and responsibility
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.
Introduction to Electrodynamics
David J. Griffiths · Edition: 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.
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.
Check your preparation
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 six-step study plan
Steps completed: 0 / 6
These are your own study marks, not an assessment. They are not sent to this site.
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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.
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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.
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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.
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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.
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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.
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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.
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.
- 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.
- Faraday’s law gives emf=−dPhi/dt=−0.030 V. The negative sign means clockwise circulation viewed from +z, opposing the increasing +z flux.
- With the stated resistive model, I=emf/R=−0.010 A. Dissipation is I²R=0.00030 W=0.30 mW, always nonnegative.
Answer and interpretation
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
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.
- 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.
- 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.
- 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.
Answer and interpretation
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.
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.
Sources and edition checks
Sources checked:
- OpenStax: calculus-based volume2 preface
The official volume provides a calculus-based entry and distinguishes original publication from current digital notices.
- Cambridge: fifth-edition front matter
Cambridge identifies fifth edition, selected hardcover and the vector-analysis/electrodynamics reading scope.
- Cambridge: Student Guide series catalogue
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.