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We know that time varying magnetic field produces voltage in a coil. But what if the wire forms a semicircle or an arc instead of a complete loop, will such magnetic field produce voltage between the ends of the wire? If yes, what is the formula to calculate the induced voltage?

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Yes, it induces the voltage across the both ends, but no current can flow. This is similar to motion of conductor inside a magnetic filed, or motional EMF. The motional EMF is described as: $$\oint ( \vec{v}^{\,}\times \vec{B}^{\,})d \vec{l}^{\,} $$ Here $\vec{v}^{\,} $ is the speed, $\vec{B}^{\,}$ is the magnetic field, and the integral should be evaluated along the conductor curve. This situation is not exactly a time varying magnetic filed, but everything here is relative if you know what i mean. The best example is antenna. The equation doesn't provide an immediate quantitive values to your question, i try here to make it clear that the EMF can be induced in an open loop circuit.

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