Rotating charged half ring

Electricity and Magnetism Level 5

A charged half ring of radius \(R\) is held fixed on a plane. Let the center of the half ring be \(O\).

Let us complete the circle \( C\) which contains the half ring. A point \(P\) is taken on the circumference of \(C\), such that \(\overrightarrow {PO}\) passes through the center of mass of the half ring.

The plane is then rotated with an angular velocity \(\omega\) about an axis passing through \(P\) and perpendicular to the plane.

Find the value of \[\dfrac {|\vec{B_P}|}{|\vec{E_P}|} \cdot 10^{16}\] where \(|\vec{B_P}|\) and \(|\vec{E_P}|\) are the magnitudes of magnetic and electric fields at point \( P\) respectively.


\(\bullet \ \ \ \) The mass distribution of the half ring is uniform.

\(\bullet \ \ \ \ \)The charge distribution of the half ring is uniform.

\(\bullet \ \ \ \ \)The above procedure is carried out in vacuum, so the permeability \(\mu_{o}=4\pi \times 10^{-7} \ \text {H/m}\)

\(\bullet \ \ \ \ \)When the plane rotates, the angular velocity of the half ring with respect to the plane is \(0\)

\(\bullet \ \ \ \ \)Coulomb's constant \(k=9 \times 10^9 \ \text {N} \text{m}^2/\text {C}^2\)

\(\bullet \ \ \ \ \omega=9 \ \text{rad/s}\)

\(\bullet \ \ \ \ R=1 \ \text{m}\)


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