Faraday’s Law Visualizer
Faraday’s Law Visualizer computes and visualizes the induced electromotive force (EMF) in a loop due to a changing magnetic field, illustrating Faraday’s Law of Induction for electromagnetism studies.
Formulas Used in Faraday’s Law Visualizer
The visualizer uses the following formulas for a loop in a changing magnetic field:
Faraday’s Law:
\\[ \mathcal{E} = -\frac{d\Phi_B}{dt} \\]Magnetic Flux:
\\[ \Phi_B = B A \cos\theta \\]Magnetic Field (Sinusoidal):
\\[ B = B_0 \sin(\omega t) \\]Induced EMF:
\\[ \mathcal{E} = -A B_0 \omega \cos(\omega t) \\]Where:
- \\( \mathcal{E} \\): Induced electromotive force (V)
- \\( \Phi_B \\): Magnetic flux (Wb)
- \\( B_0 \\): Magnetic field amplitude (T)
- \\( A \\): Loop area (m²)
- \\( \omega \\): Angular frequency (rad/s)
- \\( t \\): Time (s)
- \\( \theta = 0^\circ \\): Angle between magnetic field and loop normal (assumed perpendicular)
Example Calculations
Example 1: Standard Case
Input: B₀ = 0.1 T, A = 0.01 m², ω = 100 rad/s, t = 0 s
\\[
\Phi_B = 0.1 \cdot 0.01 \cdot \cos(0^\circ) = 0.001 \, \text{Wb}
\\]
\\[
\mathcal{E} = -0.01 \cdot 0.1 \cdot 100 \cdot \cos(100 \cdot 0) = -0.1 \, \text{V}
\\]
Result: EMF = -0.1 V at t = 0 s
Example 2: Larger Area
Input: B₀ = 0.1 T, A = 0.05 m², ω = 100 rad/s, t = 0 s
\\[
\Phi_B = 0.1 \cdot 0.05 \cdot \cos(0^\circ) = 0.005 \, \text{Wb}
\\]
\\[
\mathcal{E} = -0.05 \cdot 0.1 \cdot 100 \cdot \cos(100 \cdot 0) = -0.5 \, \text{V}
\\]
Result: EMF = -0.5 V at t = 0 s
Example 3: Higher Frequency
Input: B₀ = 0.1 T, A = 0.01 m², ω = 200 rad/s, t = 0 s
\\[
\Phi_B = 0.1 \cdot 0.01 \cdot \cos(0^\circ) = 0.001 \, \text{Wb}
\\]
\\[
\mathcal{E} = -0.01 \cdot 0.1 \cdot 200 \cdot \cos(200 \cdot 0) = -0.2 \, \text{V}
\\]
Result: EMF = -0.2 V at t = 0 s