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A closed surface consists of a cone and a circular base. The base has a radius 5.7𝑚, and the height of the cone is 5.1𝑚. There is a constant magnetic field with magnitude 2.5𝑇, that makes an angle 66∘ with the horizontal. What is the magnetic flux through the top of the cone?

Question

A closed surface consists of a cone and a circular base. The base has a radius 5.7𝑚, and the height of the cone is 5.1𝑚. There is a constant magnetic field with magnitude 2.5𝑇, that makes an angle 66∘ with the horizontal. What is the magnetic flux through the top of the cone?

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Solution

The magnetic flux through a surface is given by the equation Φ = ∫ B • dA, where B is the magnetic field, dA is a differential area vector, and the dot product implies that only the component of the field that is perpendicular to the surface contributes to the flux.

However, the magnetic field is given at an angle to the horizontal, not perpendicular to the surface of the cone. Therefore, the magnetic flux through the top of the cone is zero, because the magnetic field lines are parallel to the surface, not cutting through it.

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