IB Physics HLTopic 4 — InductionPaper 1 & 2Φ = BA cos θ~15 min read
Magnetic Flux
Last page we kept saying a conductor “cuts field lines” — a nice picture, but a bit vague. Time to make it exact. Magnetic flux is simply how much magnetic field passes through a loop of area. Hold the loop face-on to the field and you catch the maximum number of field lines. Turn it edge-on and the lines slip past without going through — zero flux. Everything about induction, Faraday’s law included, comes down to this one quantity changing.
📘 What you need to know
Magnetic flux is how much magnetic field passes perpendicularly through an area
It is the product of the flux density and the area perpendicular to the field: Φ = BA (when field and area are square-on)
Flux uses the symbol Φ (Greek “phi”) and is measured in Webers (Wb)
When the field is at an angle, only the perpendicular component counts: Φ = BA cos θ
θ is the angle between the field and the normal (the line perpendicular to the area)
Flux is maximum when field lines are perpendicular to the area (θ = 0°) and zero when parallel (θ = 90°)
Magnetic flux linkage is the flux multiplied by the number of turns: NΦ = BAN cos θ
Flux linkage is measured in Weber-turns (Wb turns)
An e.m.f. is induced whenever the flux linkage changes — through a change in B, A, or θ
What flux actually means
Picture the magnetic field as a set of parallel lines and a loop of wire as a hoop held up in that field. Magnetic flux is the number of field lines that thread through the hoop. The more lines caught, the more flux.
Magnetic flux (field square-on to the area)Φ = BAΦ = flux (Wb) • B = flux density (T) • A = area (m2)
How many lines you catch depends entirely on how you hold the loop. Two extremes make the point:
Face-on, the loop catches every line and the flux is maximum. Edge-on, the lines skim past without threading through, so the flux is zero. Orientation is everything.
A weber sounds exotic but it’s just a tesla times a square metre: 1 Wb = 1 T m2. So flux is nothing more than “field strength × the area it passes through”. If you ever forget the units, rebuild them from Φ = BA and you can’t go wrong.
When the field is at an angle
Most of the time the field won’t hit the loop dead-on. When it arrives at a slant, only the part of the field that is perpendicular to the area actually threads through. We measure the tilt using the normal — an imaginary line sticking straight out of the surface — and take the cosine.
Magnetic flux at an angleΦ = BA cos θθ = angle between the field and the normal to the area (degrees)
The angle θ is measured from the normal, not the surface. When the field lines up with the normal (θ = 0°), cos 0 = 1 and the flux is maximum.
The cosine does exactly what the picture demands:
Angle to normal θ
Field lines are…
Flux
θ = 0°
Perpendicular to the area (through it)
Maximum, Φ = BA
θ = 90°
Parallel to the area (skim past)
Zero, Φ = 0
in between
Partly through
Φ = BA cos θ
Here’s the trap that catches everyone: θ is measured from the normal, not from the surface of the coil. So a loop lying flat face-on to the field has θ = 0° (maximum flux), even though it “looks” horizontal. When in doubt, draw the little normal arrow first and measure the angle from that.
WE 1
A circular coil of radius 4.0 cm sits in a uniform magnetic field of flux density 0.80 T. (a) Calculate the maximum magnetic flux through the coil. (b) Calculate the flux when the field makes an angle of 60° with the normal to the coil. (c) State the angle between the field and the coil’s plane in part (b).
(a) Step 1 — area of the circle, then maximum fluxA = πr² = π(0.040)² = 5.03 × 10⁻³ m²Φₓₕₓ = BA = (0.80)(5.03 × 10⁻³)Φₓₕₓ = 4.0 × 10⁻³ Wb(b) Step 2 — field at 60° to the normalΦ = BA cosθ = (4.0 × 10⁻³) × cos 60°Φ = 2.0 × 10⁻³ Wb(c) angle to the plane
The plane and the normal are 90° apart.
90° − 60° = 30° to the planecos 60° = 0.5, so tilting to 60° from the normal exactly halves the flux — a clean check. And watch part (c): “angle to the normal” and “angle to the plane” always add to 90°.
Flux linkage: stacking up the loops
A real coil isn’t one loop — it’s many. Each turn of wire captures the same flux, so with N turns the total “flux experienced” is N times bigger. This is called magnetic flux linkage.
Magnetic flux linkageNΦ = BAN cos θN = number of turns • units: Weber-turns (Wb turns)
Flux Φ = BA cos θ
× number of turns
Flux linkage NΦ = BAN cos θ
Flux linkage matters because it’s the quantity that has to change to induce an e.m.f. And there are three ways to change it:
Change the field strengthB (move a magnet closer, or use an electromagnet)
Change the areaA in the field (slide a rod on rails to grow the loop)
Change the angleθ (rotate the coil — exactly what a generator does)
WE 2
A rectangular solenoid coil of 240 turns has cross-sectional dimensions 5.0 cm by 8.0 cm. It sits in a uniform field of flux density 0.15 T. (a) Calculate the flux linkage when the field is perpendicular to the plane of the coil. (b) Calculate the flux linkage when the coil is rotated so the field makes 40° with the normal.
(a) Step 1 — area, then flux linkage at θ = 0A = 0.050 × 0.080 = 4.0 × 10⁻³ m²NΦ = BAN = (0.15)(4.0 × 10⁻³)(240)NΦ = 0.14 Wb turns(b) Step 2 — rotate to 40° from the normalNΦ = BAN cosθ = 0.144 × cos 40°NΦ = 0.11 Wb turns“Perpendicular to the plane” means the field lines up with the normal, so θ = 0 and cosθ = 1 — that’s the maximum. Rotating away from that only ever reduces the linkage.
WE 3
A rectangular window frame is 30 cm wide and 50 cm tall. When closed, its plane is perpendicular to the Earth’s magnetic field of flux density 2.0 × 10−5 T. (a) Calculate the flux through the frame when closed. (b) The window is opened until its plane makes 60° with the field. Calculate the new flux. (c) Sketch how the flux varies as the window is opened from closed (0°) round to 180°.
(a) Step 1 — closed: field along the normalA = 0.30 × 0.50 = 0.15 m²Φ = BA = (2.0 × 10⁻⁵)(0.15)Φ = 3.0 × 10⁻⁶ Wb (3.0 μWb)(b) plane at 60° to the field → normal at 30°Φ = BA cos 30° = (3.0 × 10⁻⁶) × 0.866Φ = 2.6 × 10⁻⁶ Wb(c) the shape of the grapha cosine curve: max at 0°, zero at 90°, then negative to a minimum at 180°Careful with part (b): the question gives the angle to the plane (60°), so the angle to the normal is 90 − 60 = 30°. Slot 30° into cosθ, not 60°. This plane-versus-normal swap is the single most common slip in flux questions.
As the coil rotates, the flux traces a smooth cosine: greatest when face-on, zero when edge-on at 90°, then swinging negative as the far face turns toward the field. That changing flux is what drives a generator.
🧲 Working a flux question
Find the area in m2. Square? A = side². Circle? A = πr2 (from the data booklet).
Convert units: cm → m, mT → T. Area errors are the number-one mistake.
Field square-on?Φ = BA. At an angle? Φ = BA cos θ.
Check the angle.θ is measured from the normal. Given the angle to the plane? Use 90° − that.
Coil with N turns? Multiply by N for flux linkage: NΦ = BAN cos θ.
Units: flux in Wb, flux linkage in Wb turns.
💡 Top tips
The angle θ is always measured from the normal — draw the normal arrow first.
Given the angle to the plane? Convert: θ from normal = 90° − (angle to plane).
Flux is max when field lines are perpendicular to the area (θ = 0°), zero when parallel (θ = 90°).
Get the area in m2 first — convert cm to m before squaring.
Circle area is A = πr2; other shapes are in the data booklet.
Flux is in Wb; flux linkage NΦ is in Wb turns — don’t mix them up.
⚠ Common mistakes
Measuring θ from the surface instead of the normal — they differ by 90°
Using cos when the angle given is to the plane (you need 90° − it first)
Leaving the area in cm2 — 1 cm² = 10−4 m², not 10−2
Squaring before converting: do cm → m first, then square
Forgetting the N when the question asks for flux linkage
Thinking a flat, face-on loop has zero flux because it “looks horizontal” — it’s actually maximum
Confusing flux (Wb) with flux linkage (Wb turns)
Quick recap:Magnetic fluxΦ = BA is how much field passes through an area, measured in Webers. When the field is tilted, only the perpendicular part counts: Φ = BA cos θ, where θ is the angle from the normal. Flux is maximum at θ = 0° and zero at θ = 90°. For a coil of N turns, the flux linkage is NΦ = BAN cos θ, in Weber-turns. An e.m.f. appears whenever this linkage changes — through a change in B, A, or θ.
Now we have the missing piece. Last page an e.m.f. came from a conductor “cutting field lines”; this page we’ve defined exactly what’s being cut — flux linkage. Put the two together and you get the single most important law in this whole topic: the induced e.m.f. equals the rate of change of flux linkage. That’s Faraday’s law, and it’s next. Next page: Faraday’s Law of Induction.
Flux and flux linkage getting tangled?
Book a free meeting and we’ll sort out Φ = BA cos θ, the normal-versus-plane angle, and flux linkage for coils.
