Difficulty: Intermediate | Prerequisites: Understanding of magnetic fields (B-fields), vector dot products, basic calculus (derivatives and integrals), familiarity with SI units for voltage, magnetic field strength, and area.
Faraday's Law connects two ideas you have already met separately: magnetic fields and electric circuits. It says that a changing magnetic environment near a conductor can push charges around a loop, producing a voltage (EMF) without a battery. This is the principle behind electric generators, transformers, and induction cooktops. If you are comfortable with magnetic fields and the dot product, this lab applies both in a hands-on setting.
A voltage appears in a closed loop whenever the magnetic flux through that loop changes over time. The faster or larger that change, the bigger the voltage. Increasing the number of loops (turns) multiplies the effect proportionally.
Magnetic flux (Φ_B)
The total amount of magnetic field passing through a given area. Calculated as the integral of the magnetic field dotted with the area vector: Φ_B = ∫ B · dA. In simple terms, think of it as "how much magnetic field threads through your loop." More field, a bigger loop, or better alignment all mean more flux.
Electromotive force (EMF, ε)
The induced voltage in a circuit caused by a changing magnetic flux. Despite its name, it is not a force; it is measured in volts. Think of it as the "push" that drives current around a loop when the magnetic environment changes.
Faraday's Law
The relationship ε = −N (dΦ_B / dt). It states that the induced EMF equals the negative of the number of turns times the rate of change of magnetic flux. In simple terms, this means: change the flux quickly, get a bigger voltage; add more loops, get a bigger voltage.
Number of turns (N)
The count of loops in a coil. Each loop contributes to the total induced EMF, so doubling the turns doubles the voltage for the same rate of flux change.
High gain offset
A small, constant voltage reading present in the iOLab's high gain sensor even when no flux is changing. It must be measured (from a flat region of the graph) and subtracted from your peak readings to get the true induced EMF.
Magnetometer
A sensor (located under the "M" symbol on the iOLab) that measures the local magnetic field in three orthogonal directions (Bx, By, Bz). Requires calibration before use.
The induced EMF in a coil of N turns is:
ε = −N (dΦ_B / dt)
The magnetic flux through one loop is:
Φ_B = ∫ B · dA
For a uniform field and a flat loop, this simplifies to:
Φ_B = B · A · cos θ
where θ is the angle between the magnetic field vector and the area vector (the normal to the loop's surface).
The negative sign comes from Lenz's Law: the induced EMF opposes the change in flux that created it.
Three quantities sit inside Φ_B = B · A · cos θ. Changing any one of them changes the flux and therefore induces an EMF:
Change B: Strengthen or weaken the magnetic field passing through the loop.
Change A: Expand or shrink the area of the loop (less common in simple experiments).
Change θ: Rotate the loop relative to the field direction. This is the method used in this lab when working with Earth's field, which has a fixed magnitude at a given location.
A larger peak flux change produces a larger EMF, but so does making that change happen faster. Both matter.
Rotating a multi-turn coil quickly through 180° in Earth's magnetic field is the standard strategy in this lab, because you cannot change B (Earth's field is fixed) or easily change A mid-experiment.
Quantity | Formula | Units |
|---|---|---|
Magnetic flux | Φ_B = ∫ B · dA = B A cos θ (uniform field) | Weber (Wb) = T·m² |
Induced EMF | ε = −N (dΦ_B / dt) | Volts (V) |
Earth's magnetic field (typical) | ~25–65 μT depending on location | Microtesla (μT) |
Key relationship to remember: every variable you can increase – N, B, A, or the speed of the flux change – will raise the induced EMF.
Faraday's Law is the operating principle of every electrical generator and transformer. Power stations spin coils inside magnetic fields to convert mechanical energy into the alternating current that reaches your wall socket. On a smaller scale, the wireless charging pad for a phone uses a changing magnetic flux to induce a voltage in a coil inside the device.
"EMF is a force." It is not. EMF is a voltage (measured in volts). The name is historical and misleading.
"A strong magnetic field automatically means a large induced voltage." No. The field must be changing through the loop. A constant, very strong field through a stationary loop induces zero EMF.
"The sign of the EMF on the graph matters for reporting magnitude." In this lab, only the absolute value of the peak voltage matters. The sign tells you the direction of current flow, which is not being assessed here.
"More wire always means more voltage." Only if the extra wire adds complete turns through which flux changes. Extra wire that does not enclose additional area just adds resistance without increasing EMF.
⚠️ Be able to state Faraday's Law in equation form and identify each variable.
⚠️ Know the three ways to change magnetic flux (change B, change A, change θ) and be able to give an example of each.
⚠️ Understand why the negative sign exists (Lenz's Law: the induced current opposes the change).
⚠️ Be able to explain why rotating a coil in a uniform field induces an EMF, even though B is constant.
True or false: A stationary loop in a constant magnetic field will have an induced EMF. (False)
Fill in the blank: Doubling the number of turns in a coil while keeping everything else the same will ______ the induced EMF. (double)
True or false: EMF is measured in newtons. (False, it is measured in volts.)
Fill in the blank: The SI unit of magnetic flux is the ______. (weber, Wb)
True or false: Rotating a coil so that θ changes from 0° to 180° in Earth's field will induce a voltage. (True)
Q: Write Faraday's Law and define every symbol.
A: ε = −N (dΦ_B / dt). ε is the induced EMF (volts), N is the number of turns in the coil, Φ_B is the magnetic flux through one turn (webers), and t is time (seconds).
Q: A single circular loop of area 0.01 m² sits in a uniform magnetic field of 50 μT, with the field perpendicular to the loop. What is the magnetic flux through the loop?
A: Φ_B = B · A · cos 0° = (50 × 10⁻⁶ T)(0.01 m²)(1) = 5 × 10⁻⁷ Wb = 0.5 μWb.
Q: Explain why spinning a coil faster in Earth's field increases the measured EMF.
A: Spinning faster reduces the time interval dt over which the flux changes. Since ε = −N (dΦ_B / dt), a smaller dt for the same total ΔΦ_B gives a larger magnitude of EMF.
Q: A student measures a peak voltage of 0.45 mV and a high gain offset of 0.02 mV. What is the corrected induced EMF?
A: 0.45 mV − 0.02 mV = 0.43 mV.
Q: Why does Lenz's Law require the negative sign in Faraday's Law?
A: The induced EMF drives a current whose own magnetic field opposes the change in flux that caused it. This is a consequence of conservation of energy: if the induced current reinforced the change, you would get energy from nothing.
This connects to Lenz's Law (the direction/sign of induced EMF), which is often tested as a separate concept but is embedded in the negative sign of Faraday's Law. It also links to AC generators and transformers in later chapters, where a continuously rotating coil in a magnetic field produces sinusoidal voltage. Understanding magnetic flux here also reinforces the idea of field line counting used in Gauss's Law for electric fields.
Faraday's Law, electromagnetic induction, induced EMF, magnetic flux, magnetic flux density, weber, tesla, Lenz's Law, rate of change of flux, number of turns, coil, solenoid, AC generator, transformer, iOLab high gain sensor, Earth's magnetic field, PHYS 142, Physics 142 Lab 7, electromotive force