AcademyElectromagnetic Induction
Academy
Faraday's Law
Level 1 - Physics topic page in Electromagnetic Induction.
Principle
Faraday's law makes induced emf equal to the negative rate of change of magnetic flux linkage.
Notation
\(\mathcal E\)
induced emf
\(\mathrm{V}\)
\(N\)
number of turns in the coil
1
\(\Phi_B\)
magnetic flux through one turn
\(\mathrm{Wb}\)
\(B\)
uniform magnetic field magnitude
\(\mathrm{T}\)
\(A\)
loop area
\(\mathrm{m^{2}}\)
\(\theta\)
angle between field and area normal
\(\mathrm{rad}\)
Method
Derivation 1: Flux linkage
A coil with \(N\) identical turns links the same flux through each turn.
One-turn flux
\[\Phi_B=\int\vec B\cdot d\vec A\]
Flux linkage
\[N\Phi_B\]
Derivation 2: Faraday's law
The induced emf depends on how fast the flux linkage changes. The minus sign encodes the opposition direction.
Induction law
\[\mathcal E=-\frac{d}{dt}(N\Phi_B)\]
Constant turns
\[\mathcal E=-N\frac{d\Phi_B}{dt}\]
Magnitude
\[|\mathcal E|=N\left|\frac{d\Phi_B}{dt}\right|\]
Derivation 3: Uniform-field loop
If \(B\), \(A\), or \(\theta\) changes, differentiate \(BA\cos\theta\) with respect to time.
Uniform flux
\[\Phi_B=BA\cos\theta\]
Changing field only
\[\mathcal E=-NA\cos\theta\frac{dB}{dt}\]
Changing area only
\[\mathcal E=-NB\cos\theta\frac{dA}{dt}\]
Rules
These are the working forms of Faraday's law.
Faraday's law
\[\mathcal E=-N\frac{d\Phi_B}{dt}\]
Magnitude
\[|\mathcal E|=N\left|\frac{\Delta\Phi_B}{\Delta t}\right|\]
Uniform flux
\[\Phi_B=BA\cos\theta\]
Changing field
\[\mathcal E=-NA\cos\theta\frac{dB}{dt}\]
Examples
Question
A
\[30\]
-turn coil has flux per turn changing from \[0.020,mathrm{Wb}\]
to \[0.005,mathrm{Wb}\]
in \[0.10,mathrm{s}\]
Find \[|mathcal E|\]
Answer
[|mathcal E|=Nleft| rac{DeltaPhi_B}{Delta t}
ight|=30 rac{|0.005-0.020|}{0.10}=4.5,mathrm{V}]
Checks
- Use flux per turn with the factor \(N\), not total area unless the turns are distinct surfaces.
- The minus sign gives direction; magnitudes use absolute values.
- Flux is in webers, and \(1\,\mathrm{Wb\,s^{-1}}=1\,\mathrm{V}\).
- If \(B\), \(A\), and \(\theta\) are all constant, \(\mathcal E=0\).