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While coming across electromagnetic induction, I learned that for an emf to be induced in a coil of wire there should be a change in magnetic flux which is given by $\frac{BA\cos\theta}{t}$ where theta is angle between vector normal to the plane of coil and field lines vector.

But in the case of a single straight wire (not carrying any current) that has a cylindrical shape what should be the vector normal to plane of wire? Can we still apply $BA\cos\theta $ for magetic flux? If yes what should be the theta?

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In your formula theta is the angle between the magnetic field direction at the wire’s location and the direction that is normal (perpendicular) to the plane of one turn in the coil. For example if the coil is a stack of wire circles then the coil’s direction is normal to the plane of any one circle.

Faraday’s law applies to a region of space whether a conducting coil is present or not. Changing magnetic flux through the region induces an electric field that circulates around the boundary of the region. If a conductor happens to be present then that electric field induces current to flow in the conductor.

In the scenario you describe, the conducting cylinder is really just a single circular loop whose normal is directed along the length of the cylinder. The angle you want is the angle between that and the magnetic field. The cylinder must be straight or the angle is not well defined. Also, any induced current will flow in circles, not along the wire’s length.

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