Electromagnetic Induction & Faraday's Law Simulator
Set coil parameters and magnetic field for real-time visualization of magnetic flux Φ, induced EMF ε, and induced current I. Supports Lenz law animation, mutual inductance, and transformer calculations.
Parameters
Mode
Coil settings
Turns N
Cross-sectional area A
cm²
Resistance R
Ω
Magnetic field settings B(t) = B₀·sin(2πft)
Amplitude B₀
T
Frequency f
Hz
Mutual induction settings
N₂ (secondary turns)
Coupling coefficient k
Playback Controls
Waveform Overlay
Save: 0 / 5
Results
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ε_max inductionElectromotive force [V]
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I_max induced current [A]
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M mutual inductance [mH]
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P consumed power [W]
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N₂/N₁ transformer turns ratio
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Φ_max maximum magnetic flux [mWb]
Time-series graph
Animation（Lenz law）
Anim
CAE Applications
Preliminary estimation of motor back-EMF / Magnetic flux density evaluation for transformer core design / Wireless power transfer (WPT) coil coupling coefficient evaluation. Useful for verifying design values before FEM analysis (JMAG/Ansys Maxwell).
Theory & Key Formulas
Faraday's law (induced EMF):
$$\varepsilon = -N\frac{d\Phi}{dt}, \quad \Phi = B \cdot A \cos\theta$$
Sine wavemagnetic field $B(t)=B_0\sin(2\pi ft)$ when ：
$$\varepsilon(t) = -N A B_0 \cdot 2\pi f \cdot \cos(2\pi ft)$$
What exactly is "magnetic flux" and why does changing it create electricity?
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Basically, magnetic flux ($\Phi$) is a measure of the total magnetic field passing through a loop, like our coil. Think of it as the number of magnetic field lines going through the coil's area. Faraday discovered that a change in this flux—whether the field strength changes, the coil moves, or it rotates—induces a voltage (EMF) in the coil. In this simulator, you change the flux by adjusting the Amplitude B₀ or Frequency f of the oscillating magnet.
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Wait, really? So the negative sign in Faraday's law... what's that about?
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That's Lenz's Law in action! The negative sign means the induced EMF (and thus the current it drives) creates its own magnetic field that opposes the change that caused it. It's nature's way of resisting change. For instance, if you increase B₀ with the slider, the induced current will try to create a field to reduce that increase. Try it—watch the direction of the induced current (I₂) flip when the magnetic flux graph starts decreasing.
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Okay, I see the primary coil making flux. What's the point of the N₂ (Secondary Turns) and Coupling Coefficient k sliders?
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Great question! Now you're modeling a transformer. N₂ is the number of turns in a second, separate coil. The Coupling Coefficient k (from 0 to 1) represents how well the magnetic flux from the first coil links to the second. If k=1 (perfect coupling), all flux from coil 1 goes through coil 2. If k=0, they're not linked at all. Slide k and watch the induced EMF in the secondary coil ($\varepsilon_2$) change. This is the core of wireless power transfer!
Physical Model & Key Equations
The fundamental law of electromagnetic induction is Faraday's Law. The induced Electromotive Force (EMF) in a coil is proportional to the rate of change of magnetic flux through it and the number of turns.
$$\varepsilon = -N \frac{d\Phi}{dt}$$
Here, $\varepsilon$ is the induced EMF (Volts), $N$ is the number of coil turns, and $\Phi$ is the magnetic flux (Webers). The flux is the product of the magnetic field $B$, the coil area $A$, and the cosine of the angle $\theta$ between the field and the area vector: $\Phi = B \cdot A \cos\theta$.
In this simulator, the magnetic field oscillates sinusoidally: $B(t) = B_0 \sin(2\pi f t)$. Plugging this into Faraday's Law gives us the specific formula for the induced EMF over time.
$$\varepsilon(t) = -N A B_0 \cdot 2\pi f \cdot \cos(2\pi f t)$$
Here, $B_0$ is the magnetic field amplitude (set by the **Amplitude B₀** slider), $f$ is the frequency, and $A$ is the cross-sectional area of the coil. The result shows that the induced EMF is also a sine wave, but it's a *cosine*, meaning it's 90 degrees out of phase with the magnetic field—a direct consequence of the derivative in Faraday's Law.
Frequently Asked Questions
Increasing the amplitude (B₀) of the magnetic field proportionally increases the maximum induced electromotive force. Raising the frequency f also increases the amplitude of the electromotive force in proportion to the frequency, as can be seen from Faraday's law equation ε(t) = -N A B₀·2πf·cos(2πft). It can also be observed that the waveform period becomes shorter and the changes become steeper.
Increasing the number of turns N increases the induced electromotive force ε in proportion to N (ε = -N dΦ/dt). However, since the coil resistance also increases almost proportionally to the number of turns, the induced current I = ε/R does not necessarily scale proportionally. In this simulator, the resistance value can be set separately, so you can experiment with the effects of both factors.
The animation visualizes the direction of the magnetic field (counter-field) generated by the induced current using arrows. When the external magnetic field increases, the counter-field is generated in a direction that opposes it, and when it decreases, the counter-field is generated in a direction that supplements it. This provides an intuitive understanding of Lenz's law, which states that the induced current always flows in a direction that opposes the change in magnetic flux.
You can set the number of turns, voltage, and frequency of the primary coil, the number of turns of the secondary coil, and the coupling coefficient (strength of mutual inductance). Based on these, the induced voltage on the secondary side is automatically calculated from the turns ratio and displayed in real time along with the time variation of the magnetic flux. Setting the coupling coefficient to less than 1 also allows the effect of leakage flux to be reproduced.
Real-World Applications
Electric Motors & Generators: This is Faraday's Law in motion. In a generator, a coil is rotated in a magnetic field (changing $\theta$), inducing an AC voltage. Conversely, in a motor, applying a voltage causes motion. The "back-EMF" in a motor is the voltage induced by its own rotation, opposing the applied voltage—exactly what you see with the negative sign in the simulator.
Transformers: By adjusting the **N₂** and **k** sliders, you are simulating a transformer. AC in the primary coil creates a changing flux, which induces a voltage in the secondary coil. The ratio of turns ($N_2/N_1$) determines if the voltage is stepped up or down. This is how wall adapters convert 120V AC to lower voltages for your devices.
Wireless Power Transfer (WPT): This is a transformer with a low coupling coefficient (k < 1), like your wireless phone charger. The primary and secondary coils are not physically connected. The simulator shows how crucial maximizing 'k' and the operating frequency 'f' are for efficient energy transfer over an air gap.
CAE & Engineering Design: Before running complex Finite Element Method (FEM) simulations in tools like Ansys Maxwell or JMAG, engineers use these fundamental equations for a "sanity check" or preliminary design. Estimating back-EMF in a motor or the coupling between coils in a transformer ensures the FEM model is set up correctly and yields plausible results.
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 "if the magnetic flux changes, a current *must* flow," but if the circuit is not closed, no current will flow. An induced electromotive force (EMF) is a state where a "voltage" is generated. It's like an electrical outlet with the switch off—you can't extract energy from it alone. For example, if the secondary side of a transformer is left open (unconnected), a voltage is generated, but the current is zero. This illustrates the difference between "EMF" and "current."
Next, regarding parameter settings, you should be careful to keep realistic orders of magnitude in mind. If you set extreme values just for fun, like a "Number of turns N" to 10,000 or a "Frequency f" to 1 MHz, the calculation might show a huge voltage, but in reality, parasitic capacitance in the coil and heat generation would prevent such performance. In practical design, for a small transformer, the number of turns is typically in the hundreds, and frequencies are usually 50/60 Hz or in the range of several kHz to several hundred kHz. The correct approach is to first get the theoretical value with this tool, and then consider real-world constraints (wire gauge, core saturation, losses).
Also, do you think "a coupling coefficient k=1" means the highest efficiency? It's true in an ideal sense, but in real magnetic coupling, k=1 is virtually impossible; even achieving above 0.95 requires extremely high-precision design. In wireless charging, it's not uncommon for k to drop below 0.7 due to coil misalignment. Try lowering k from 1 to 0.5 in this tool. You'll see the voltage induced on the secondary side drop significantly. This is your first step in understanding the "gap between theory and implementation."