The induction of a uniform magnetic field inside a cylinder of radius
$r$ = 0.1 m increases linearly with time:
$B = \alpha t$ (coefficient
$\alpha$ = 10
$^{-3}$ T/s). The magnetic field is directed along the axis of the cylinder. What is the strength of the eddy electric field at a distance of
$l$ = 0.2 m from the cylinder axis?
<h3 id="back-link"><a href="/#11.2">$\leftarrow$Back</a></h3>
<h3> Statement </h3>
<p>
$11.2.2$
The induction of a uniform magnetic field inside a cylinder of radius $r$ = 0.1 m increases linearly with time: $B = \alpha t$ (coefficient $\alpha$ = 10$^{-3}$ T/s). The magnetic field is directed along the axis of the cylinder. What is the strength of the eddy electric field at a distance of $l$ = 0.2 m from the cylinder axis?
</p>
<h3>Solution</h3>
<p>
Magnetic field passes through an area of $\pi r^2$ m$^2$ and its flux increases with time. So, this causes that appears a rotational electric field around cylinder axis such that its induced magnetic field opposes to $\vec{B}(t)$ (Lenz Law). Applying Faraday's Law,
$$\oint\vec{E}\cdot \vec{ds} = \frac{d\Phi_B}{dt}$$
this closed integral is for concentric circular paths about cylinder's axis. Then, for a distance $a$ from center,
$$E(a) \cdot 2\pi a = \pi r^2 \alpha$$
$$E(a) = \frac{\alpha r^2}{2a}$$
Finally, evaluating for $a=l$,
$$E = \frac{\alpha r^2}{2l}$$
</p>
<h4>Answer</h4>
<p>
$$E = 2.5\~\cdot\~10^{-5}\~{\rm{\frac{V}{m}}}$$
</p>
<p style="text-align: right; font-style: italic; font-size: 14;">
BSc. Luis Daniel Fernández Quintana<br>
Physics Department (FCNE)<br>
Universidad de Oriente, Cuba<br>
</p>
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