Adjust coil turns, cross-sectional area, peak flux density, and frequency to visualize the 90° phase relationship between B(t) and induced EMF(t) via Faraday's Law in real time.
Coil Parameters
Turns N
Coil area A
cm²
Peak flux density B_max
T
Frequency f
Hz
Results
While paused, move the sliders to update the result instantly.
Live values (magnet passing through coil)
0.00
Induced EMF ε (V)
0.00
Flux rate dΦ/dt (Wb/s)
0.00
Flux linkage NΦ (mWb)
0.00
Induced current I (mA)
—
Peak EMF (V)
—
RMS EMF (V)
Magnet passing through coil — Faraday's & Lenz's Law
N pole (magnet)S poleInduced currentFlux through coil
Faster magnet motion means a larger rate of flux change and a larger EMF. When the magnet stops, the EMF drops to zero.
Reference: sinusoidal B(t) and EMF(t) (90° phase shift)
What exactly is the "phase shift" this simulator shows between the magnetic field and the induced voltage?
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Basically, it's a timing difference. The magnetic field $B(t)$ changes first, and the induced electromotive force (EMF) $\varepsilon$ reacts to that rate of change. In this simulator, when $B(t)$ is a sine wave, the induced EMF becomes a cosine wave. Try moving the Frequency (f) slider up and down. You'll see the waves squeeze or stretch, but the EMF peak always occurs when the magnetic field is crossing zero—that's the 90-degree phase shift.
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Wait, really? So the voltage isn't highest when the magnet is strongest? Why does the number of coil turns (N) make the voltage bigger?
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Right! The voltage is highest when the field is changing the fastest, which is when it passes through zero. Each loop of the coil acts like a separate "voltage source." More turns (N) means you're linking more of the changing magnetic flux, so you add up more of those tiny induced voltages. For instance, in a power transformer, thousands of turns are used to step up voltage. Try increasing the Turns (N) parameter in the simulator and watch the amplitude of the red EMF wave grow dramatically.
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That makes sense. What about the coil area (A) and the peak flux (B_max)? They seem to do similar things in the equation.
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Great observation. They both increase the total magnetic flux $\Phi$ going through the coil. A bigger area (A) captures more field lines. A stronger peak field (B_max) means more field lines are there to be captured. In practice, you might increase A by making a bigger coil, or increase B_max by using a stronger magnet. A common case is in an electric guitar pickup: a strong magnet (high B_max) and many turns of fine wire (high N) are used to get a strong signal from the vibrating string. Play with both sliders—you'll see they multiply together to scale the EMF.
Physical Model & Key Equations
The core principle is Faraday's Law of Induction. A changing magnetic flux through a loop of wire induces an electromotive force (EMF, or voltage). The magnetic field in this simulator varies sinusoidally over time.
$$B(t) = B_{\max}\sin(2\pi f t)$$
Where $B_{\max}$ is the peak magnetic flux density (Tesla), $f$ is the frequency of oscillation (Hz), and $t$ is time (seconds).
The total magnetic flux $\Phi$ through a coil of $N$ turns, each with area $A$, is the product. The induced EMF $\varepsilon$ is the negative rate of change of this flux.
$$
\Phi(t) = N \cdot A \cdot B(t) \quad \Rightarrow \quad \varepsilon = -\frac{d\Phi}{dt}= -NAB_{\max}2\pi f\cos(2\pi ft)
$$
The negative sign represents Lenz's Law: the induced EMF creates a current whose magnetic field opposes the change that produced it. The result shows the EMF is proportional to $N$, $A$, $B_{\max}$, and $f$, and is 90 degrees out of phase (a cosine) with the magnetic field (a sine).
Frequently Asked Questions
According to Faraday's law, the induced electromotive force is proportional to the time derivative of the magnetic flux. The derivative of a sine wave is a cosine wave, which is 90 degrees ahead in phase relative to the original waveform, resulting in a 90-degree phase difference between the magnetic flux waveform and the EMF waveform.
Increasing the number of turns N or the cross-sectional area A proportionally increases the total magnetic flux Φ, thereby increasing the amplitude of the induced electromotive force. The shape and phase difference of the waveform remain unchanged, and only the amplitude changes, which can be observed in real time.
From Faraday's law equation ε = -N A B_max 2πf cos(2πft), a higher frequency f results in a greater rate of change of magnetic flux over time, causing the amplitude of the induced electromotive force to increase proportionally. The period of the waveform becomes shorter.
It is useful for intuitively understanding physical phenomena and the relationships between parameters, but it does not account for non-ideal effects such as actual coil resistance, leakage flux, or core saturation. Please use it as a qualitative learning tool.
Real-World Applications
Electric Generators (Alternators): This simulator directly models the core of a generator. A rotating magnet (creating a changing B-field) inside a stationary coil induces an AC voltage. The frequency (f) is determined by the rotation speed (RPM), and the output voltage is controlled by the magnet strength, coil size, and number of turns.
Transformers: Transformers use two coils wrapped around a common iron core. An alternating current in the primary coil creates a changing magnetic flux (B_field) in the core. This changing flux induces a voltage in the secondary coil, stepping the voltage up or down depending on the ratio of turns (N) between the coils.
Induction Cooking: A cooktop contains a coil of wire (with high N and driven at high frequency f) that generates a rapidly alternating magnetic field. This field induces eddy currents in the metal pot placed above it, and the resistance of the pot to these currents generates heat directly in the cookware.
Magnetic Flow Meters: Used in pipelines to measure the flow rate of conductive fluids. A magnetic field (B) is applied across the pipe, and the moving fluid acts like a moving conductor. The motion induces a voltage (EMF) perpendicular to both the flow and the field, which is measured and is directly proportional to flow velocity.
Common Misconceptions and Points to Note
When you start using this simulator, there are a few points that are easy to misunderstand. First, you might think "a stronger magnet always yields a larger voltage," but that's only half true. While increasing the maximum magnetic flux density $B_{\max}$ does increase the EMF, the rate of change of the magnetic flux is critically important. For example, no matter how strong a magnet you hold stationary against a coil, the voltage will be zero. Conversely, even a weak magnet can generate a significant voltage if moved rapidly. When you increase the "Frequency f" in the simulator and see the EMF surge, that's precisely this "rate of change" effect in action.
Next, regarding the displayed negative sign. The minus in the formula $\varepsilon(t) = -\frac{d\Phi}{dt}$ simply indicates "direction." It only reverses the polarity you'd measure with a voltmeter and doesn't affect the "magnitude" of the voltage lighting a bulb. Therefore, when considering the maximum EMF value, you can usually ignore this minus and think in terms of absolute value. However, be careful: this sign becomes crucial when connecting multiple coils or in circuit design where you need to track the precise current direction.
Finally, note that the coil's cross-sectional area A is fixed. In real-world design, if you try to increase the number of turns N, you're forced to use thinner wire to fit within the same space. This increases the wire resistance, leading to more heat, and the actual power you can extract from that nice high voltage decreases. The simulator shows ideal conditions, so increasing turns always seems beneficial, but in practice, you must constantly consider the trade-off between number of turns, wire gauge, and coil size.
