# Magnetic Flux Linkage ## What is Flux Linkage? - Flux linkage is particularly useful when dealing with coils or solenoids that have multiple loops of wire - Rather than considering each individual loop separately, flux linkage combines the effect of all turns together

Flux linkage is the product of magnetic flux and the number of turns in the coil

## The Flux Linkage Equation - The basic equation for flux linkage is: $$\text{Flux linkage} = N\Phi = BAN$$ - Where: - $N$ = number of turns in the coil - $\Phi$ = magnetic flux (Wb) - $B$ = magnetic flux density (T) - $A$ = cross-sectional area (m²) - Flux linkage has units of **Weber turns (Wb turns)** ## Why Flux Linkage Matters - When you have a coil with many turns, each loop experiences the magnetic flux - The total effect is multiplicative - doubling the number of turns doubles the flux linkage - This is why transformers and generators use coils with hundreds or thousands of turns - More turns means a greater flux linkage for the same magnetic field ## Non-Perpendicular Flux Linkage - Just like with simple magnetic flux, the angle between the field lines and the coil matters - When the field isn't perfectly perpendicular to the coil's face, we need to account for the angle ### The Angle-Dependent Equation - For flux linkage at an angle: $$N\Phi = BAN\cos\theta$$ - Where: - $\theta$ = angle between the magnetic field lines and the normal (perpendicular line) to the coil's plane

Flux linkage is maximum when $\theta = 0°$ (field perpendicular to coil) and zero when $\theta = 90°$ (field parallel to coil)

### Understanding the Angle - The normal line is perpendicular to the surface of the coil - When the normal and field lines are parallel ($\theta = 0°$), flux linkage is maximum - When the normal and field lines are perpendicular ($\theta = 90°$), flux linkage is zero ### How Flux Linkage Varies with Angle The chart below shows how flux linkage decreases as the angle between the field and the normal increases:

- The relationship follows a cosine curve - Flux linkage drops most rapidly between 60° and 90° - At 45°, you retain about 70% of the maximum flux linkage | Angle $\theta$ | $\cos\theta$ | Flux Linkage | Configuration | |---|---|---|---| | $0°$ | 1 | Maximum ($BAN$) | Field perpendicular to coil face | | $45°$ | 0.707 | $0.707 \times BAN$ | Intermediate angle | | $90°$ | 0 | Zero | Field parallel to coil face | ## Worked Example **Question:** A cylindrical solenoid has 450 turns and a circular cross-section with radius 0.25 m. It sits perpendicular to a magnetic field of flux density 6.8 mT. Calculate the flux linkage through this solenoid. **Solution:** **Step 1: Identify the known values** - Number of turns, $N = 450$ turns - Radius, $r = 0.25$ m - Magnetic flux density, $B = 6.8$ mT $= 6.8 \times 10^{-3}$ T **Step 2: Calculate the cross-sectional area** Since the solenoid has a circular cross-section: $$A = \pi r^2 = \pi \times (0.25)^2 = 0.196 \text{ m}^2$$ **Step 3: Write the flux linkage equation** Since the field is perpendicular to the coil: $$N\Phi = BAN$$ **Step 4: Substitute values and calculate** $$N\Phi = (6.8 \times 10^{-3}) \times 0.196 \times 450$$ $$N\Phi = 0.599 \text{ Wb turns} = 0.60 \text{ Wb turns (2 s.f.)}$$ ## Key Differences from Simple Flux | Magnetic Flux ($\Phi$) | Flux Linkage ($N\Phi$) | |---|---| | For a single loop | For multiple loops | | Units: Weber (Wb) | Units: Weber turns (Wb turns) | | $\Phi = BA\cos\theta$ | $N\Phi = BAN\cos\theta$ | | Doesn't account for number of turns | Multiplies by number of turns | ## Practical Considerations - In electromagnetic induction, it's the **change** in flux linkage that matters most - Devices like transformers and generators rely on changing flux linkage to induce voltages - A coil with 100 turns in the same field has 100 times the flux linkage of a single loop - This is why electromagnets and inductors are always made with many turns of wire

Don't confuse the angle $\theta$ - it's measured from the normal to the coil, not from the coil face itself

## Tips for Calculations - Always check whether the angle is given or if the field is perpendicular (meaning you can use the simpler equation without $\cos\theta$) - Remember to convert units: mT to T, cm to m, etc. - When calculating area for circular coils, use $A = \pi r^2$ - For rectangular coils, use $A = \text{length} \times \text{width}$ - The number of turns $N$ is always a dimensionless number (no units)