11.2.2 | revision #12595 | Savchenko Solutions

Statement

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?

Solution

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 $$a=l$$ ,

$E = \frac{\alpha r^2}{2l}$

Answer

$E = 2.5~\cdot~10^{-5}~{\rm{\frac{V}{m}}}$