EMI BYU 14 - CHATUP707

Solution to Problem 14

Solution: Force on a Vibrating Coil

1. Relation between EMF and Flux Gradient

When the coil vibrates with amplitude $a$ and frequency $f$, its position is $z = a \sin(\omega t)$ and velocity is $v = a\omega \cos(\omega t)$. The induced EMF is given by: $$ \varepsilon = \left| \frac{d\Phi}{dt} \right| = \left| \frac{d\Phi}{dz} \frac{dz}{dt} \right| = \frac{d\Phi}{dz} v $$ The amplitude of voltage $V_0$ corresponds to maximum velocity $v_{max} = a\omega$: $$ V_0 = \frac{d\Phi}{dz} (a\omega) \implies \frac{d\Phi}{dz} = \frac{V_0}{a\omega} $$

2. Calculation of Force

The magnetic force on a coil carrying current $I$ in a non-uniform field is $F = I \frac{d\Phi}{dz}$ similar to Electric Dipole where it is $F = p \frac{dE}{dz}$ . Substituting the derived gradient: $$ F = I \frac{V_0}{a (2\pi f)} $$

Substituting Values:
$I = 0.2 \, \text{A}$, $V_0 = 5\pi \, \text{V}$, $a = 10^{-3} \, \text{m}$, $f = 100 \, \text{Hz}$.

        $$ F = \frac{0.2 \times 5\pi}{10^{-3} \times 2\pi \times 100} = \frac{\pi}{0.2\pi} = 5.0 \, \text{N} $$
    