13.3 Motional emf  (Page 4/6)

A rectangular coil rotating in a magnetic field

A rectangular coil of area A and N turns is placed in a uniform magnetic field $\overrightarrow{B}=B\widehat{j},$ as shown in [link] . The coil is rotated about the z -axis through its center at a constant angular velocity $\omega .$ Obtain an expression for the induced emf in the coil.

Strategy

According to the diagram, the angle between the perpendicular to the surface ( $\widehat{n}$ ) and the magnetic field $(\overrightarrow{B})$ is $\theta$ . The dot product of $\overrightarrow{B}·\widehat{n}$ simplifies to only the $\text{cos}\theta$ component of the magnetic field, namely where the magnetic field projects onto the unit area vector $\widehat{n}$ . The magnitude of the magnetic field and the area of the loop are fixed over time, which makes the integration simplify quickly. The induced emf is written out using Faraday’s law.

Solution

When the coil is in a position such that its normal vector $\widehat{n}$ makes an angle $\theta$ with the magnetic field $\overrightarrow{B},$ the magnetic flux through a single turn of the coil is

From Faraday’s law, the emf induced in the coil is

The constant angular velocity is $\omega =d\theta \text{/}dt.$ The angle $\theta$ represents the time evolution of the angular velocity or $\omega t$ . This is changes the function to time space rather than $\theta$ . The induced emf therefore varies sinusoidally with time according to

where ${\epsilon }_0=NBA\omega .$

Significance

If the magnetic field strength or area of the loop were also changing over time, these variables wouldn’t be able to be pulled out of the time derivative to simply the solution as shown. This example is the basis for an electric generator, as we will give a full discussion in Applications of Newton’s Law .

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