This cheat sheet covers magnetic flux and Lenz's law, two key ideas in electromagnetic induction. Students use these ideas to predict when a changing magnetic field creates an induced current. Worked examples help connect formulas to direction rules, sign conventions, and real circuit behavior.
This reference is useful for reviewing before quizzes, labs, and AP or upper high school physics exams.
The central formula for magnetic flux is , where is the angle between the magnetic field and the area vector. Faraday's law gives induced emf as , and the negative sign represents Lenz's law. Lenz's law says the induced current creates a magnetic field that opposes the change in flux.
Most problems require identifying what changes, calculating flux or emf, then using the right-hand rule to find current direction.
Key Facts
- Magnetic flux through a flat loop is , where is magnetic field strength, is area, and is measured from the area vector.
- The SI unit of magnetic flux is the weber, with .
- Faraday's law for a coil with turns is for average induced emf.
- For changing field strength with constant area and angle, the induced emf magnitude is .
- For a rotating loop, the flux can be written as when the area vector rotates with angular speed .
- Lenz's law says the induced current flows in the direction that creates a magnetic field opposing the change in magnetic flux.
- If magnetic flux into the page increases, the induced current produces a field out of the page, so the current is counterclockwise.
- If the loop is part of a circuit with resistance , the induced current magnitude is .
Vocabulary
- Magnetic Flux
- Magnetic flux is the amount of magnetic field passing through a surface, calculated by for a uniform field.
- Area Vector
- The area vector is a vector perpendicular to a surface with magnitude equal to the surface area.
- Induced EMF
- Induced emf is the voltage produced when magnetic flux through a loop changes over time.
- Faraday's Law
- Faraday's law states that the induced emf in a coil is .
- Lenz's Law
- Lenz's law states that an induced current creates a magnetic field that opposes the change in flux that produced it.
- Right-Hand Rule
- The right-hand rule relates current direction to magnetic field direction by curling the fingers with current so the thumb points along the loop's field.
Common Mistakes to Avoid
- Using the angle between the magnetic field and the surface is wrong because in is measured between and the area vector, not the surface plane.
- Forgetting the number of turns is wrong because a coil multiplies the induced emf by , so the correct relationship is .
- Treating the negative sign as a negative voltage only is wrong because the minus sign in Faraday's law represents Lenz's law and gives the opposition direction.
- Assuming any magnetic field creates current is wrong because induction requires changing flux, so a constant , constant , and constant produce .
- Choosing current direction before identifying the flux change is wrong because Lenz's law depends on whether flux into or out of the page is increasing or decreasing.
Practice Questions
- 1 A single circular loop has radius and is perpendicular to a uniform field of . What is the magnetic flux through the loop?
- 2 A coil with turns has area . The magnetic field perpendicular to the coil changes from to in . What is the magnitude of the induced emf?
- 3 A loop with resistance has an induced emf of . What is the induced current magnitude?
- 4 A magnetic field into the page through a loop is decreasing. Use Lenz's law and the right-hand rule to determine whether the induced current is clockwise or counterclockwise, and explain why.
Understanding Magnetic Flux and Lenz's Law Worked Examples
A useful way to think about flux is as a count of how much magnetic field passes through a chosen surface. The surface has two possible directions, like the two sides of a sheet of paper. Choosing one direction sets the positive direction for the problem.
This choice is not a physical change in the loop. It is a bookkeeping choice. Once chosen, keep it for every step.
A field pointing along the chosen surface direction gives positive flux. A field pointing opposite that direction gives negative flux.
Many sign errors happen when students confuse the plane of a loop with its area vector. The area vector is always perpendicular to the plane, not along it.
Induction depends on a change over time, not simply on the presence of a magnetic field. A stationary loop in a strong, steady field has no induced emf. A loop can have zero flux yet still have an induced emf if its flux is changing at that instant.
For example, when a loop rotates, it may be edge-on to the field for a moment. Its flux is then zero, but the flux can be changing most rapidly. This is why generators use rotating coils.
Mechanical motion continually changes the orientation of a coil relative to a magnetic field, producing an alternating voltage. The faster the rotation, the faster the flux changes and the larger the voltage can become.
For Lenz's law, state the change before deciding on a current direction. Do not start by guessing clockwise or counterclockwise. First decide whether the original flux is increasing or decreasing.
Next decide which induced field would resist that change. If flux into the page is decreasing, the induced field must point into the page because it tries to maintain the disappearing flux. The right hand rule then gives the current direction.
Curl your fingers in the current direction and your thumb shows the field through the loop. This method works even when the external field points out of the page or when a magnet moves away instead of toward the coil.
The opposition in Lenz's law is required by energy conservation. If an approaching magnet made a loop produce a field that pulled the magnet closer, the magnet could speed up while electrical energy appeared in the circuit without an energy source. Instead, the induced field resists the approach.
Someone moving the magnet must do work. That work becomes electrical energy, then often thermal energy in the wire or a resistor. Similar effects appear in bicycle dynamos, wireless charging pads, induction cooktops, metal detectors, transformer circuits, and magnetic braking systems.
In worked problems, list known quantities, convert every value to standard units, find the initial and final flux separately, then divide the flux change by the time interval. Use the magnitude for current size. Use Lenz's law separately for direction.