Motional Emf using Faraday's Law

Claimed by Chelsea Calhoun

The Main Idea

When a wire moves through an area of magnetic field, a current begins to flow along the wire as a result of magnetic forces. Originally, we learned to calculate the motional emf in a moving bar by using the equation [math]\displaystyle{ {\frac{q(\vec{v} \times \vec{B})L}{q}} }[/math]. However, there's an easier way to do this: by writing an equation for emf in terms of magnetic flux.

A Mathematical Model

We know that the magnitude of motional emf is equal to the rate of change of magnetic flux. [math]\displaystyle{ |emf| = \left|\frac{d\Phi_m}{dt}\right| }[/math]

We also know that magnetic flux is defined by the formula: [math]\displaystyle{ \Phi_m = \int\! \vec{B} \cdot\vec{n}dA }[/math]

When a metal bar or wire is moving across an area, creating a closed circuit, the magnetic field contained inside the area of the circuit is no longer constant. When there is a non-constant magnetic field, we must use Faraday's law (a combination of the two equations above) to solve for motional emf.

Faraday's Law is defined as: [math]\displaystyle{ emf = \int\! \vec{E} \cdot d\vec{l} = -\frac{d}{dt} \int\! \vec{B} \cdot \vec{n}dA }[/math]

where [math]\displaystyle{ \vec{E} }[/math] is the electric field along the path, [math]\displaystyle{ l }[/math] is the length of the path you're integrating on, [math]\displaystyle{ \vec{B} }[/math] is the magnetic field inside the area enclosed, and [math]\displaystyle{ \vec{n} }[/math] is the unit vector perpendicular to area A.

A Computational Model

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