Faradays Law Of Induction

Why your transformer gets warm and stays warm

The voltage across a coil doesn't come from the magnet. It comes from the rate at which the magnetic field through that coil changes. That's Faradays Law Of Induction in its simplest form. The formula you'll see everywhere is EMF = -N(d/dt). N is the number of turns. is magnetic flux. The negative sign is Lenz's law, meaning the induced voltage always opposes the change that created it. That's not philosophy. That's conservation of energy showing up as a sign convention. People miss the practical part. Flux isn't just "magnet strength." Flux is B times A times cos(theta), where theta is the angle between the field lines and the normal to your coil surface. Rotate the coil 90 degrees and your flux goes to zero even though the magnet hasn't moved at all. Rotate it back quickly and you get a spike. Change the area by collapsing the coil or stretching it, and you get voltage. Move the magnet, move the coil, change the field strength with an electromagnet, or change the geometry. Any of these produce EMF. The equation doesn't care which mechanism you use.

Calculating induced voltage in a practical coil setup

Here's how I actually do it when I'm on a bench and someone hands me a problem. First, draw the coil and label the area. Not the coil area. The projected area perpendicular to the expected flux direction. Then figure out how B changes over the time interval you care about. If the magnet is moving at constant velocity through the coil, B isn't linear with time because the field falls off with distance cubed for a dipole. So d/dt isn't constant either. The voltage pulse looks like a differentiated bell curve, not a square wave. Measuring it with a scope will show a positive peak as the magnet enters, zero at center, and a negative peak as it leaves. The integral of that whole waveform is zero if the magnet starts and ends outside the coil. That's a useful sanity check. If your measurement doesn't integrate to roughly zero, something is wrong with your probe grounding or your coil isn't symmetric. For a solenoid with N turns, length l, and core area A carrying a time-varying current, the self-induced voltage is L(di/dt) where L = mu*N^2*A/l for an air core. Swap in a ferrite core and multiply mu by the relative permeability, but only up to saturation. Ferrite saturates around 0.3 to 0.5 tesla depending on the grade. Beyond that, L drops sharply and your inductor stops behaving like an inductor. It becomes a resistor with a weird phase angle. I once spent three days debugging a pickup coil that kept producing half the voltage I calculated. The math was right. The coil had 470 turns of 36 AWG on a ferrite rod core. I was driving a small electromagnet at 50 Hz and expecting a clean sine wave output. Instead, the peaks were clipped. Flat on top. I assumed my function generator was distorting. I swapped it. Same result. I measured the core flux with a search coil and a ballistic galvanometer and found the core was saturating at about 0.35 T. The ferrite was a generic MZ-2000 type rated for power transformers, not high-permeability small-signal work. I switched to a fair-rite 43 mix toroid, dropped the drive current by a third, and got the expected 1.8 volts RMS instead of the clipped 0.9. The calculation hadn't been wrong. The core permeability assumption had been wrong. Mu_r isn't a fixed number. It's a function of B, frequency, and temperature. Datasheets give you a curve, not a single value. I stopped treating it like one.

What most people get wrong about mutual inductance

Mutual inductance M between two coils isn't just N1*N2*something. The coupling coefficient k can range from near zero to nearly one depending on geometry, placement, and core presence. Two identical coils placed one centimeter apart on the same axis with air cores might have k around 0.1. Put a high-permeability core through both and align them carefully and you might reach 0.8 or 0.9. The formula M = k*sqrt(L1*L2) is correct but k is the hard part to predict without simulation or measurement. You can't derive it from first principles easily for arbitrary geometries. Finite element software like FEMM or ANSYS Maxwell will give you a good answer in ten minutes. Analytical approximations only work for simple cases like coaxial solenoids with infinite length assumptions that don't match reality. Another thing that trips people up. The induced EMF in a stationary loop due to a changing magnetic field is a non-conservative electric field. You can't assign a scalar potential to it the way you do with electrostatic fields. If you try to analyze a circuit with a transformer using nodal analysis and treat the secondary as a regular voltage source, you'll get the right numbers for steady state but you'll misunderstand what's happening. The secondary voltage isn't a source. It's an induced field. The current that flows depends on the load, and that current creates its own flux that opposes the primary flux. That's the feedback loop that makes transformers work as impedance transformers and why a transformer with an open secondary still draws magnetizing current from the primary.

When Faraday's law breaks down or needs correction

The basic law assumes the loop is stationary and the field is changing, or the loop moves through a static field. In conductors, you also have to account for the motional EMF component separately when doing a full relativistic treatment. For most low-frequency engineering work, combining them into the single flux derivative is fine. At high frequencies, skin effect changes the effective area of your conductor. A thick copper wire carrying high-frequency induced current only uses the outer shell. Your effective cross-section shrinks, resistance goes up, and your measured voltage drops because the coil heats and the wire resistivity increases with temperature. Copper's temperature coefficient is about 0.00393 per degree Celsius. A coil that reads 1 ohm at room temperature will be 1.27 ohms at 70 degrees C. That's a 27 percent increase in loss, not a negligible amount if you're building a sensitive pickup or a current transformer. Another failure mode. If your magnetic field changes so fast that the induced electric field creates currents which then create opposing fields that cancel the change before it fully penetrates the conductor, you're in the regime where eddy currents dominate. This is exactly what happens in induction cooktops and metal detectors. The skin depth delta = sqrt(2/(omega*mu*sigma)) tells you how deep the field penetrates. For copper at 100 kHz, delta is about 0.2 millimeters. A 2 millimeter copper plate effectively blocks most of the field. If you're designing a sensor and your target material is conductive, you're not measuring the target's magnetic properties. You're measuring eddy current reflections. The signal you see is dominated by conductivity and geometry, not permeability. Beginners often blame their coil design when the problem is the target material. If you need a quick reference or calculation tool, search for the NIST database on magnetic properties or use open-source tools like FEMM for 2D field modeling. The law itself is simple. The applications are where the complexity lives.