7. EMF in Vibrating Wire
Step 1: Wire Equation
The wire of length $l$ vibrating in its fundamental mode has the displacement equation:
$$ y(x,t) = a \sin\left(\frac{\pi x}{l}\right) \sin(2\pi \nu t) $$The transverse velocity is $v_y = \frac{\partial y}{\partial t}$:
$$ v_y = 2\pi \nu a \sin\left(\frac{\pi x}{l}\right) \cos(2\pi \nu t) $$Step 2: Motional EMF Integral
The motional EMF in a segment $dx$ is $d\mathcal{E} = v_y B dx$. Total EMF is the integral over the length.
$$ \mathcal{E} = \int_{0}^{l} v_y B \, dx = B (2\pi \nu a \cos(2\pi \nu t)) \int_{0}^{l} \sin\left(\frac{\pi x}{l}\right) dx $$Step 3: Solving the Integral
$$ \int_{0}^{l} \sin\left(\frac{\pi x}{l}\right) dx = \left[ -\frac{l}{\pi} \cos\left(\frac{\pi x}{l}\right) \right]_0^l = -\frac{l}{\pi} (-1 – 1) = \frac{2l}{\pi} $$Substitute back:
$$ \mathcal{E} = B (2\pi \nu a \cos(2\pi \nu t)) \left( \frac{2l}{\pi} \right) $$
$$ \mathcal{E} = 4 a \nu B l \cos(2\pi \nu t) $$