Magnetic Flux
Magnetic flux Φ_B is the signed surface integral of B⃗ through an oriented area. For a uniform field across a flat surface it reduces to Φ_B=B⃗·A⃗=BA cosθ, where θ is measured from the area normal—not from the plane. Flux is the quantity whose change drives electromagnetic induction, so sign, area-vector orientation, and the difference between open-surface flux and closed-surface net flux must be controlled before using Faraday’s law.
Topic sequence · 1 of 6
Learning objectives
Objective 1
Define the area vector, determine the sign of magnetic flux from B⃗·A⃗, and calculate Φ_B for uniform fields and flat surfaces without confusing the normal angle with the plane angle.
Objective 2
Set up and evaluate Φ_B=∫B⃗·dA⃗ for nonuniform fields or surfaces, using symmetry, coordinates, and correct differential area elements.
Objective 3
Connect magnetic flux to Gauss’s law for magnetism and prepare for Faraday’s law by identifying every way flux can change: field magnitude, area, orientation, or the portion of a loop overlapping a field region.
- Dot products and vector components, especially the interpretation of cosθ as selecting the component normal to a surface.
- Electric flux from Unit 8, including oriented area elements and surface integrals.
- Magnetic-field sources and directions from Unit 12, including nonuniform fields near wires and coils.
- Gauss’s law for magnetism from Topic 12.1: net magnetic flux through every closed surface is zero.
Definitions first
Representations you must read and create
Representation 1
Flat loop in uniform B⃗: draw the surface plane, the area vector normal, the field vector, and θ between B⃗ and the normal. Avoid labeling the complementary angle to the plane as θ unless you explicitly convert it.
Representation 2
Flux-versus-angle graph: Φ_B=BA cosθ is maximum at 0°, zero at 90°, negative for angles greater than 90°, and returns to −BA at 180°. The sign records orientation, not “negative field strength.”
Representation 3
Nonuniform-field heat map over a rectangular surface: divide the area into strips or elements, identify B_n(x,y), and integrate rather than multiplying one representative B by the total area.
Representation 4
Closed surface in a dipole field: field lines can enter and leave different faces, giving nonzero face-by-face fluxes while the total outward sum is zero.
Representation 5
Loop moving into a finite magnetic-field region: the relevant area is the overlapping area, not necessarily the loop’s full geometric area. This representation prepares the dA/dt reasoning in Topic 13.2.
Representation 6
Flux-versus-time graph: before calculating induced emf, identify slope sign and magnitude. Faraday’s law responds to dΦ_B/dt, not directly to the value of Φ_B.
Core relationships and calculus
Variables: Φ_B Wb; B T; A m²; θ angle between B⃗ and A⃗
Use: Use for a uniform magnetic field across a flat surface. θ=0 gives maximum positive flux; θ=90° gives zero flux.
Variables: Φ_B Wb; B T; dA m²
Use: Use for nonuniform fields, curved surfaces, or any situation where the normal component changes across the area.
Variables: dΦ_B Wb; B_n T; dA m²
Use: Local form: integrate the signed normal component B_n over the surface.
Variables: B T; dA m²
Use: Gauss’s law for magnetism. Net magnetic flux through a closed surface is always zero in the course model.
Variables: Φ_B Wb; B T; A_perp m²
Use: Equivalent geometric view for uniform B: A_perp is the projected area perpendicular to the field, including sign through the chosen normal orientation.
A surface integral is a limit of a signed sum: Φ_B=lim Σ(B⃗_i·n̂_i)ΔA_i. For a flat surface with constant normal, this becomes ∫B_n dA. If B varies only with x across a rectangle of height h, use dA=h dx. If the surface is curved, both B and the normal direction can vary, so parameterize the surface or exploit a projection/symmetry argument. The calculus is not optional decoration: it is what makes “field through area” precise for nonuniform situations.
For a uniform field across a flat area, every differential area element has the same normal n̂ and the same B⃗. Then Φ_B=∫B⃗·n̂ dA=(B cosθ)∫dA=BA cosθ. This shows why the simple dot-product expression is a special case of the surface integral, not a separate law. It also shows why θ must be the angle between B⃗ and the normal: the dot product selects the normal component.
Magnetic flux is not a count of literal field lines and it is not a vector. It is a signed scalar tied to a chosen surface orientation. Do not import electric Gauss-law source logic into an open magnetic-flux calculation: a loop can have nonzero magnetic flux even though no magnetic monopole exists. The zero-net-flux rule applies only to a closed surface. Topic 13.1 defines and calculates flux; induced emf, Lenz’s law, and induced electric fields belong to Topic 13.2.
Physical examples
Example 1
A 0.020 m² loop in a 0.50 T uniform field has Φ_B=0.010 Wb when its area vector is parallel to B, zero flux when its plane is perpendicular to the area vector and the normal is 90° to B, and −0.010 Wb when the chosen normal is reversed.
Example 2
If a problem says the magnetic field is 30° above the plane of a loop, then the angle with the area vector is 60°. The flux magnitude is BA cos60°=BA/2, not BA cos30°.
Example 3
A rectangular loop spans a field B(x)=B₀+kx perpendicular to its plane. Slice the loop into strips dA=h dx and integrate h∫B(x)dx across its width.
Example 4
A curved surface can have changing local normals even when B is uniform. The surface integral must follow B⃗·dA⃗ locally unless a projection or closed-surface shortcut applies.
Example 5
A bar magnet passing near a loop can change flux even if the loop’s area and orientation remain fixed, because both the field magnitude and its spatial distribution through the loop change.
Example 6
A loop rotating at constant angular speed in uniform B has time-dependent flux BA cosθ(t). Topic 13.2 will differentiate that flux to obtain a sinusoidal induced emf.
Worked reasoning
Exam reasoning patterns
Reasoning pattern 1
Draw the area vector first. If the problem gives an angle to the plane, convert it to the angle with the normal before using cosθ.
Reasoning pattern 2
Use the sign of B⃗·A⃗, not a memorized “into page is negative” rule. Into/out of page becomes positive or negative only after you choose the area-vector direction.
Reasoning pattern 3
When B varies across the surface, write the integral with correct limits and dA. A single BA value is unjustified unless B_n is constant or you have explicitly used an average justified by the field profile.
Reasoning pattern 4
For a closed surface, individual face fluxes may be nonzero. Set the algebraic sum to zero rather than setting every face flux to zero.
Reasoning pattern 5
For a partially overlapping field region, use the area actually threaded by B. The full loop area is irrelevant where B=0.
Reasoning pattern 6
Before moving to Faraday’s law, sketch Φ_B(t). The induced-emf sign and magnitude depend on the time derivative of flux, so the flux graph is often the fastest intermediate representation.
Deeper calculus and limiting-case check
Suppose a circular disk of radius R lies in the xy plane and experiences B_z(r)=B₀(1−r²/R²). With dA=2πr dr, Φ_B=2πB₀∫₀ᴿ(r−r³/R²)dr=2πB₀[R²/2−R²/4]=(πB₀R²)/2. The field at the center is B₀, but the average normal field over the disk is B₀/2. This example makes clear why “center field × area” fails for a nonuniform field and provides an immediate average-value check.
Interactive magnetic-flux integral check
Compare uniform-field flux BA cosθ with a nonuniform rectangular-field integral B(x)=B₀(1+αx), where the field is normal to the rectangle.
Experimental connection
Map magnetic flux experimentally with a loop/coil and a magnetic-field sensor. One option is to measure B at a grid of points across a planar loop, estimate Φ_B numerically with ΣB_nΔA, and compare with a simpler uniform-field approximation. Another is to rotate a fixed-area coil in an approximately uniform field and record B, orientation, and projected area. State the chosen area-vector direction, include sensor offset/background subtraction, propagate or discuss area/angle uncertainty, and compare measured flux trends with BA cosθ rather than relying only on a single orientation.
Connections across Physics C
Connection 1
Topic 12.1 gives ∮_closedB⃗·dA⃗=0; Topic 13.1 focuses on the generally nonzero flux through an open surface used for induction problems.
Connection 2
Topic 12.3 provides nonuniform fields from wires and loops that may require a true surface integral when calculating flux through a nearby area.
Connection 3
Topic 12.4 supplies the approximately uniform interior field of a long solenoid, making solenoid-coil flux calculations especially clean.
Connection 4
Topic 13.2 differentiates magnetic flux with respect to time. Flux is therefore the bridge between magnetic-field geometry and induced emf.
Connection 5
Electric flux in Unit 8 uses the same dot-product and oriented-area mathematics, so the representation skill transfers directly even though the source laws differ.
Related lessons and resources
Use these links when the current model depends on a prerequisite or becomes the foundation for the next model.
Common mistakes
Avoid this
Using the angle between B⃗ and the plane in BA cosθ instead of the angle between B⃗ and the area normal.
Avoid this
Treating flux as a vector or as always positive. Flux is a signed scalar set by the dot product and chosen orientation.
Avoid this
Assuming a closed surface has zero magnetic field because its net magnetic flux is zero.
Avoid this
Using B at one point times total area when B varies substantially across the surface.
Avoid this
Forgetting that reversing the area vector reverses the flux sign but does not change the physical magnetic field.
Avoid this
Using the full geometric loop area when only part of the loop overlaps the magnetic-field region.
Avoid this
Confusing a large flux value with a large induced emf; induction depends on how fast flux changes, not on flux magnitude alone.
Mastery check
- Define the main quantity without relying on an equation.
- Choose the correct representation and sign convention.
- Explain why the calculus relationship matches the physics.
- Design or interpret an experiment that could test the model.
Frequently asked questions
What is magnetic flux?
It is the signed surface integral Φ_B=∫B⃗·dA⃗, measured in webers. It quantifies the normal magnetic field threaded through an oriented surface.
What angle goes into BA cosθ?
The angle between B⃗ and the area vector, which is perpendicular to the surface.
Can magnetic flux be negative?
Yes. Negative flux means the normal component of B points opposite the chosen area vector.
Why is net magnetic flux through a closed surface zero?
Because the course model has no isolated magnetic monopoles; magnetic field lines do not begin or end inside the surface.
When do I need the integral form?
Whenever B magnitude/direction or the surface normal changes across the area, unless symmetry or projection reduces the integral.
How does magnetic flux connect to induction?
Faraday’s law states that induced emf is proportional to the negative time rate of change of magnetic flux, so Topic 13.1 supplies the quantity differentiated in Topic 13.2.