AcademyElectromagnetic Induction
Academy
Induced Electric Fields
Level 1 - Physics topic page in Electromagnetic Induction.
Principle
A changing magnetic flux creates a circulating electric field even without a conducting loop.
Notation
\(\vec E\)
induced electric field
\(\mathrm{V\,m^{-1}}\)
\(\Phi_B\)
magnetic flux through a surface
\(\mathrm{Wb}\)
\(r\)
radius of circular path
\(\mathrm{m}\)
\(R\)
radius of changing-field region
\(\mathrm{m}\)
\(B\)
uniform magnetic field in the region
\(\mathrm{T}\)
\(\mathcal E\)
circulation of electric field
\(\mathrm{V}\)
Method
Derivation 1: Faraday's law for fields
The induced emf is the line integral of \(\\vec E\) around any closed path.
emf as circulation
\[\mathcal E=\oint\vec E\cdot d\vec\ell\]
Field Faraday law
\[\oint\vec E\cdot d\vec\ell=-\frac{d\Phi_B}{dt}\]
Derivation 2: Circular path inside a uniform changing field
Symmetry makes the induced electric field tangent to circles centered on the changing magnetic field region.
Circular circulation
\[\oint\vec E\cdot d\vec\ell=E(2\pi r)\]
Flux inside path
\[\Phi_B=B\pi r^2\]
Inside field
\[E=-\frac{r}{2}\frac{dB}{dt}\]
Derivation 3: Circular path outside the changing-field region
For \(r>R\), the flux only comes from the field-filled disk of radius \(R\).
Enclosed flux
\[\Phi_B=B\pi R^2\]
Outside circulation
\[E(2\pi r)=-\pi R^2\frac{dB}{dt}\]
Outside field
\[E=-\frac{R^2}{2r}\frac{dB}{dt}\]
Rules
These results apply to circular symmetry.
Field Faraday law
\[\oint\vec E\cdot d\vec\ell=-\frac{d\Phi_B}{dt}\]
Inside region
\[E=\frac{r}{2}\left|\frac{dB}{dt}\right|\]
Outside region
\[E=\frac{R^2}{2r}\left|\frac{dB}{dt}\right|\]
Examples
Question
A uniform magnetic field inside radius
\[0.20\,\mathrm{m}\]
increases at \[3.0\,\mathrm{T\,s^{-1}}\]
Find \(E\) at \[r=0.10\,\mathrm{m}\]
Answer
\[E=\frac{r}{2}\left|\frac{dB}{dt}\right|=\frac{0.10}{2}(3.0)=0.15\,\mathrm{V\,m^{-1}}\]
Checks
- Induced electric field lines form closed loops; they do not begin or end on charge.
- The field is nonconservative when magnetic flux changes.
- The conducting wire is not the cause of the field; it only provides charges that respond.
- Direction still follows Lenz's law.