EMG-04

Faraday motional emf

ε = B ℓ v for a rod sliding on rails.

Reading speed

InductionFaraday 1831

Governing equation

\mathcal{E}=B\ell v=-\dfrac{\mathrm{d}\Phi_B}{\mathrm{d}t}

$v \in \left[0.05, 20\right]$

where

B: Field (T)
\ell: Rod length (m)
v: Speed (m/s)
\mathcal{E}: Induced emf (V)

The figure, part by part

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①

Governing law — red

\mathcal{E}=B\ell v=-\dfrac{\mathrm{d}\Phi_B}{\mathrm{d}t}

$v \in \left[0.05, 20\right]$

$v = 0.05$ $\mathcal{E} = 0.006$

Along the curve

②

Domain of the variable — blue

v_{\min} = 0.05,\quad v_{\max} = 20

$v \in \left[0.05, 20\right]$

$v = 0.05$ $v = 0.05$

Along the curve

③

Response curve — green

\mathcal{E}=B\ell v=-\dfrac{\mathrm{d}\Phi_B}{\mathrm{d}t}

$v \in \left[0.05, 20\right]$

$v = 0.05$ $\mathcal{E} = 0.006$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\mathcal{E}}{\mathrm{d}v}

$v \in \left[0.05, 20\right]$

$v = 0.175$ $\mathcal{E}^{\prime} = 0.12$

Along the curve

Lecture brief

Historical brief

Coulomb, Gauss, Ampère, Faraday and Maxwell (1861–65) unified charge, current and light. The lab computes fields, induction, Poynting flux and the electromagnetic wave in SI. This sheet (EMG-04 — Faraday motional emf) is the form associated with Faraday 1831. Working symbols: $B$ , $\ell$ , $v$ $\rightarrow$ $\mathcal{E}$ . Faraday's law ε = −dΦB/dt for a changing area ℓ x(t) in a uniform B reduces to B ℓ v.

Purpose

Purpose: compute

\mathcal{E}

from

B

\ell

v

in Electromagnetism via

\mathcal{E}=B\ell v=-\dfrac{\mathrm{d}\Phi_B}{\mathrm{d}t}

ε = B ℓ v for a rod sliding on rails. Use it when a real electromagnetism question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

B = 0.400\,\mathrm{T}

\ell = 0.300\,\mathrm{m}

v = 2.000\,\mathrm{m/s}

, the governing relation

\mathcal{E}=B\ell v=-\dfrac{\mathrm{d}\Phi_B}{\mathrm{d}t}

yields

\mathcal{E} = 0.2400\,\mathrm{V}

. Uniform B perpendicular to the loop, constant v. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Field B0.4 TRod length \ell0.3 mSpeed v2 m/s

Outputs

Induced emf \mathcal{E}0.2400 V

Reading speed

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Narration of this film

Uniform B perpendicular to the loop, constant v.

Faraday's law ε = −dΦB/dt for a changing area ℓ x(t) in a uniform B reduces to B ℓ v.

Reading speed

Watch on YouTube