AcademyElectromagnetic Induction
Academy
Lenz's Law
Level 1 - Physics topic page in Electromagnetic Induction.
Principle
Lenz's law gives the induced current direction: its magnetic effect opposes the flux change.
Notation
\(\Delta\Phi_B\)
change in magnetic flux
\(\mathrm{Wb}\)
\(\vec B_{\mathrm{ext}}\)
external magnetic field through the loop
\(\mathrm{T}\)
\(\vec B_{\mathrm{ind}}\)
field produced by the induced current
\(\mathrm{T}\)
\(I\)
induced current
\(\mathrm{A}\)
\(\mathcal E\)
induced emf
\(\mathrm{V}\)
Method
Derivation 1: State the opposition
The loop responds to the change in flux, not to the existing flux by itself.
Increasing flux
\[\frac{d\Phi_B}{dt}>0\quad\Rightarrow\quad \vec B_{\mathrm{ind}}\text{ opposes the positive flux direction}\]
Decreasing flux
\[\frac{d\Phi_B}{dt}<0\quad\Rightarrow\quad \vec B_{\mathrm{ind}}\text{ supports the original flux direction}\]
Derivation 2: Connect field direction to current direction
Once the induced field direction is known, use the right-hand rule for a current loop.
Loop field
\[\text{fingers curl with }I,\quad \text{thumb gives }\vec B_{\mathrm{ind}}\]
Direction choice
\[\vec B_{\mathrm{ind}}\text{ sets clockwise or counterclockwise current}\]
Derivation 3: Energy check
If the induced current helped the flux change, the change would reinforce itself without external work.
Opposition
\[\mathcal E=-\frac{d\Phi_B}{dt}\]
Energy consistency
\[P=I^2R\ge0\]
Rules
Use these direction rules after finding the flux change.
Faraday sign
\[\mathcal E=-\frac{d\Phi_B}{dt}\]
Resistive power
\[P=I^2R\]
Examples
Question
A magnetic field into the page through a loop is increasing. What direction is the induced current?
Answer
The induced field must point out of the page to oppose the increase into the page. By the right-hand rule, the induced current is counterclockwise.
Checks
- Oppose the flux change, not always the external field.
- First decide whether flux into or out of the page is increasing or decreasing.
- Then use the right-hand rule for the current direction.
- The induced current direction must be consistent with energy conservation.