Magnetic Flux Through a Loop Calculator
Calculate the magnetic flux through a loop from field strength (B), loop area (A) and the angle to the surface normal. Enter values in any unit and get flux in webers, millawebers and microwebers.
Find the Magnetic Flux Through a Loop
Magnetic Flux Through a Loop Formula
Where:
- \(\Phi\) — magnetic flux (webers, Wb)
- \(B\) — magnetic flux density, i.e. the strength of the uniform field (teslas, T)
- \(A\) — area enclosed by the loop (m²)
- \(\theta\) — angle between the magnetic field and the normal (perpendicular) to the loop's surface (degrees or radians)
The formula gives the total magnetic field passing through a flat loop. Only the component of the field perpendicular to the loop's surface contributes, which is why the cosine factor appears. The equivalent unit relation is \(1\ \text{Wb} = 1\ \text{T}\cdot\text{m}^2 = 1\ \text{V}\cdot\text{s}\).
Magnetic Flux Through a Loop – Calculation Example
Example: a flat circular loop with an area of 200 cm² sits in a uniform 5 mT magnetic field. The field makes a 30° angle with the normal to the loop's surface.
Convert to SI units: \(B = 5\ \text{mT} = 0.005\ \text{T}\) and \(A = 200\ \text{cm}^2 = 0.02\ \text{m}^2\).
\[ \Phi = B \cdot A \cdot \cos(\theta) = 0.005 \times 0.02 \times \cos(30^\circ) = 0.0001 \times 0.8660 \approx 8.66 \times 10^{-5}\ \text{Wb} \]
Result: \(\Phi \approx 8.66 \times 10^{-5}\) Wb, which is 0.0866 mWb or about 86.6 µWb. You can reproduce this with the calculator's default values (B = 5 mT, A = 200 cm², θ = 30°).
How to Use the Calculator
- Enter the Magnetic Flux Density (B) and select its unit (µT, mT, T or G).
- Enter the Loop Area (A) and choose the area unit (mm², cm², m², in² or ft²).
- Enter the Angle to Surface Normal (θ) in degrees, from 0 to 180. This is the angle between the field direction and the line perpendicular to the loop's surface — not the angle between the field and the plane of the loop.
- Read the Magnetic Flux result in webers, together with the same value in millawebers (mWb) and microwebers (µWb), plus a step-by-step breakdown in the calculation details.
What Is Magnetic Flux?
Magnetic flux measures how much of a magnetic field passes through a given surface, such as the area enclosed by a wire loop. Picture the field lines of a magnet: the flux counts how many of those lines cross the loop. A strong field perpendicular to the loop produces a large flux; the same field parallel to the loop's plane produces zero flux, because no field lines actually pass through it.
The angle \(\theta\) in the formula is measured from the normal to the surface — the imaginary line sticking straight out of the loop. When the field is aligned with the normal (\(\theta = 0^\circ\)), the flux is maximal and equals \(B \cdot A\). When the field lies in the plane of the loop (\(\theta = 90^\circ\)), \(\cos(90^\circ) = 0\) and no flux passes through. Angles between 90° and 180° give a negative flux, which simply means the field passes through the loop in the opposite direction.
Magnetic flux is central to electromagnetism. A changing flux through a loop induces an electromotive force (voltage) in it — this is Faraday's law of induction, the principle behind electric generators, transformers and induction cooktops. If you then need the induced voltage itself, use our Faraday's Law calculator; for the force a magnetic field exerts on a moving charge in that field, see the Magnetic Force on a Moving Charge calculator.
Interpreting the Result
The calculator returns flux in webers (Wb), the SI unit, as well as in millawebers and microwebers for convenience, since everyday flux values are often small — a 5 mT field through a 200 cm² loop yields only about 86.6 µWb. A negative result is not an error: it indicates that the field crosses the surface opposite to the chosen normal direction. If you swap the normal to the other side of the loop, the sign flips while the magnitude stays the same.
For a loop with N turns of wire (a coil), the total flux linkage is simply \(N\Phi\), since the same flux passes through each turn when the turns are tightly wound and share the same area.
Assumptions and Limitations
- Uniform field: the formula assumes the magnetic flux density \(B\) is the same everywhere over the loop. If the field varies (e.g. near a magnet's pole), the result is only an approximation based on an average value.
- Flat loop: the loop is treated as a flat (planar) surface. Strongly curved or deformable loops need a different treatment, since the angle varies across the surface.
- Angle from the normal: remember that \(\theta\) is measured from the perpendicular to the loop, not from the loop's plane. If a problem gives the angle from the plane, subtract it from 90° before entering it.
- Static situation: the calculator gives the flux at a single instant. Only a change of flux over time induces a voltage — see the Faraday's Law calculator for induced EMF.
- Units: gauss (G) is accepted for convenience (1 G = 10−4 T); all inputs are converted to SI internally.
Quick Reference: Flux vs. Angle
For \(B = 0.005\ \text{T}\) and \(A = 0.02\ \text{m}^2\) (\(B \cdot A = 10^{-4}\ \text{Wb}\)):
| θ (degrees) | cos(θ) | Flux Φ (µWb) |
|---|---|---|
| 0° | 1.000 | 100.0 |
| 30° | 0.866 | 86.6 |
| 45° | 0.707 | 70.7 |
| 60° | 0.500 | 50.0 |
| 90° | 0.000 | 0.0 |
| 120° | −0.500 | −50.0 |
| 180° | −1.000 | −100.0 |
This table shows the cosine dependence clearly: flux falls smoothly from its maximum to zero as the loop rotates from perpendicular to parallel, then becomes negative as it turns past 90°. Related field calculations you may find useful include the Electric Field of a Point Charge calculator and the Parallel Plate Capacitor calculator.
Frequently Asked Questions
How do I calculate magnetic flux by hand without a calculator?
Multiply the field strength by the loop area, then by the cosine of the angle between the field and the loop's normal: \( \Phi = B \cdot A \cdot \cos(\theta) \). First convert everything to SI units — teslas for B, square metres for A — then multiply. For example, a 0.02 T field through a 0.05 m² loop perpendicular to the field gives \( \Phi = 0.02 \times 0.05 \times 1 = 0.001 \) Wb. Keep your calculator in degree mode if the angle is given in degrees.
What is the angle if the problem gives it from the plane of the loop instead of the normal?
The formula needs the angle measured from the normal (the line perpendicular to the loop's surface). If your problem states the angle between the field and the plane of the loop, subtract it from 90° before entering it. For instance, a field at 60° to the loop's plane is at 30° to the normal, so you enter 30°. Mixing up these two conventions is the single most common mistake in flux problems and can change the answer completely.
How do I convert gauss, centimetres squared and other units before calculating?
Common conversions to SI: 1 gauss (G) = \(10^{-4}\) T, 1 mT = \(10^{-3}\) T, 1 µT = \(10^{-6}\) T; 1 cm² = \(10^{-4}\) m², 1 mm² = \(10^{-6}\) m², 1 in² = 6.4516×\(10^{-4}\) m², 1 ft² = 0.0929 m². The calculator handles these conversions automatically, but if working by hand always convert B to teslas and A to square metres first — mixing units is the fastest way to get a result that is off by several orders of magnitude.
Why is my flux negative, and is that a problem?
A negative result simply means the field passes through the loop from the opposite side relative to your chosen normal direction. It occurs whenever the angle is between 90° and 180°, making \(\cos(\theta)\) negative. It is physically meaningful: if you flip the loop over or define the normal on the other side, the sign flips while the magnitude stays the same. In Faraday's law calculations, the sign matters because it sets the direction of the induced EMF via Lenz's rule.
What is a typical value of magnetic flux, and what does one weber represent?
Everyday fluxes are tiny: Earth's magnetic field (about 50 µT) through a 100 cm² loop gives only 0.5 µWb. One weber is a large unit — it is the flux of a 1 T field through a full square metre, comparable to the flux in powerful industrial magnets. The weber equals 1 T·m² and also 1 V·s, which is why a flux changing by 1 Wb per second induces 1 volt. If you work in CGS units, note that 1 maxwell (the CGS flux unit) is only \(10^{-8}\) Wb.
Does the number of turns in a coil change the flux?
The flux through a single loop is unchanged, but a coil with N tightly wound turns has a total flux linkage of \(N\Phi\), since essentially the same flux passes through every turn. This is what matters for Faraday's law: the induced EMF is \(\varepsilon = -N \frac{d\Phi}{dt} \). So for a 200-turn coil with \(\Phi = 86.6\) µWb through each turn, the flux linkage is about 17.3 mWb. Calculate \(\Phi\) for one turn here, then multiply by N.
Can I use this calculator for a non-uniform field or a curved loop?
Not exactly. The formula \( \Phi = B \cdot A \cdot \cos(\theta) \) assumes a flat loop in a perfectly uniform field. If the field varies across the loop (for example near a magnet's pole), you can enter an average field value, but the result is only an approximation — the exact answer requires integrating B over the surface. For curved or irregularly shaped loops, the angle and field strength differ across the surface, so the same limitation applies. For a flat loop in a non-uniform field, using the average B is often good enough for estimates.