Faraday’s law explains how changing magnetic flux creates an induced electromotive force, or EMF, in a wire loop or coil. This cheat sheet helps students connect flux changes, coil motion, magnetic field strength, and circuit orientation. It is especially useful for worked examples involving changing area, changing field, and rotating generators.
Students need these tools to solve induction problems clearly and choose the correct sign and direction.
Key Facts
- Magnetic flux through a flat loop is , where is the angle between the magnetic field and the area vector.
- Faraday’s law for one loop is , and for identical turns it is .
- Lenz’s law says the induced current flows in the direction that creates a magnetic field opposing the change in magnetic flux.
- If the magnetic field changes while area and angle stay constant, the induced EMF magnitude is .
- If the loop area changes while field and angle stay constant, the induced EMF magnitude is .
- For a straight conductor of length moving at speed perpendicular to a magnetic field, the motional EMF is .
- For a rotating coil generator with turns, area , magnetic field , and angular speed , the flux is and the induced EMF is .
- The maximum generator EMF is , which increases with more turns, stronger magnetic field, larger area, or faster rotation.
Vocabulary
- Magnetic flux
- Magnetic flux is the amount of magnetic field passing through a surface, calculated by for a uniform field.
- Induced EMF
- Induced EMF is the voltage produced when the magnetic flux through a circuit changes.
- Faraday’s law
- Faraday’s law states that the induced EMF equals the negative rate of change of magnetic flux, .
- Lenz’s law
- Lenz’s law states that induced current opposes the change in magnetic flux that caused it.
- Area vector
- The area vector is a vector perpendicular to a loop’s surface, used to measure the angle in .
- Generator
- A generator converts mechanical rotation into electrical energy by using changing magnetic flux to induce EMF.
Common Mistakes to Avoid
- Using instead of for flux is wrong because is measured between the magnetic field and the area vector, so .
- Forgetting the factor gives an EMF that is too small because each turn of the coil contributes to the total induced EMF, so .
- Treating the negative sign as a negative voltage only is wrong because the minus sign in Faraday’s law represents Lenz’s law and the opposition to flux change.
- Using the final flux instead of the change in flux is wrong because induction depends on , not just at one instant.
- Ignoring units can hide errors because flux is measured in webers, where , and EMF is measured in volts.
Practice Questions
- 1 A coil has turns and area . A perpendicular magnetic field changes from to in . Find the magnitude of the induced EMF.
- 2 A metal rod of length moves at perpendicular to a magnetic field of . Calculate the motional EMF .
- 3 A generator coil has , , , and angular speed . Find .
- 4 A loop is pulled out of a region where the magnetic field points into the page. Explain, using Lenz’s law, whether the induced current is clockwise or counterclockwise.
Understanding Faraday's Law & Induced EMF Worked Examples
The most important idea in induction is that a loop responds to change, not simply to the presence of a magnetic field. A loop resting in a strong steady field can have no induced voltage. A weak field that changes rapidly can produce a noticeable voltage.
This distinction prevents a common error in worked examples. First identify what changes during the stated time interval.
It may be the field strength, the loop size, the loop orientation, or the part of the loop that lies inside a field region. If none of these changes, the induced EMF is zero.
Direction problems become easier when handled in two separate steps. Begin by deciding whether the external flux through the loop is increasing or decreasing. Then use Lenz’s law to find the direction of the magnetic field made by the induced current.
Only after that should the right hand grip rule be used to convert the induced field direction into clockwise or anticlockwise current. For example, if an external field into the page becomes stronger, the loop produces a field out of the page to resist that increase.
The current must then be anticlockwise. Keep track of the chosen surface direction throughout a calculation, since a reversed choice reverses the sign but not the physical prediction.
The negative sign in Faraday’s law represents energy conservation. An induced current creates effects that resist the motion or field change that produced it. When a magnet is pushed toward a closed conducting loop, the magnet feels a magnetic push back.
Work must be done to keep it moving. That work becomes electrical energy, and often thermal energy in the wire.
This is why a generator needs mechanical input from a turbine, hand crank, wind rotor, or engine. It cannot produce useful electrical energy without an energy source driving the rotation.
For calculations, use a clear sequence. Convert every quantity into SI units before substituting values. Area must be in square metres, time in seconds, field strength in tesla, and speed in metres per second.
Find the initial flux and final flux when more than one quantity changes. Subtract them consistently, divide by the time interval, then multiply by the number of turns. Report the magnitude unless a signed convention is requested.
In rotating coil questions, the voltage is not constant. It rises to a maximum when the flux is changing most quickly, and it is zero when the flux is momentarily greatest or smallest. This matters in alternating current generators, household power systems, bicycle dynamos, induction cookers, transformers, and wireless charging devices.
A final unit check helps. EMF is measured in volts, which is the same unit as electric potential difference.