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Question 7.4: A metal disk of radius a rotates with angular velocity ω abo...

A metal disk of radius a rotates with angular velocity ω about a vertical axis, through a uniform field B, pointing up. A circuit is made by connecting one end of a resistor to the axle and the other end to a sliding contact, which touches the outer edge of the disk (Fig. 7.15). Find the current in the resistor.

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The speed of a point on the disk at a distance s from the axis is v = ωs, so the force per unit charge is f_{mag}=v\times B =\omega sB\hat{s}. The emf is therefore

\varepsilon=\int_{0}^{a}{f_{mag}ds}=\omega B \int_{0}^{a}{sds}=\frac{\omega Ba^{2}}{2},

and the current is

I=\frac{\varepsilon }{R} =\frac{\omega Ba^{2}}{2R}.

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