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Magnetic Field at the Centre of a Ring

A ring of radius \(R\) carries a linear charge density \(\lambda\). It is rotating with angular speed \(\omega\). What is the magnetic field at the centre?

Discussion:

Linear charge density $$ \lambda=\frac{Q}{2\pi R}
$$
When the ring is rotated about the axis, the motion of the electrons in a circular orbit is equivalent to a current carrying loop.
Current $$ I=\frac{Q}{T}=\frac{Q\omega}{2\pi}$$
since Time period \(T=2\pi/\omega\).
Now, magnetic field around the centre of a current carrying loop is given by $$ B=\mu_0I/2R$$
Putting the value of \(I\) in the above equation, we get
$$ B=\frac{\mu_0\omega}{2}.\frac{Q}{2\pi R}
$$$$ \Rightarrow B=\frac{\mu_0\lambda\omega}{2}
$$

October 30, 2017

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