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(B) The magnetic field lines originate from the North pole and terminate at the South pole of a bar magnet. In the given figure,the South pole of the

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Negative

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(B) The magnetic field lines originate from the North pole and terminate at the South pole of a bar magnet. In the given figure,the South pole of the bar magnet is facing the open end of the cylinder. Therefore,the magnetic field lines are directed towards the South pole,meaning they enter the cylinder through the open end. By convention,the area vector $d\vec A$ for a closed surface is taken to be pointing outwards. Since the magnetic field $\vec B$ is entering the surface and the area vector $d\vec A$ is pointing outwards,the angle between them is obtuse (greater than $90^{\circ}$). Consequently,the dot product $\vec B \cdot d\vec A = B \, dA \cos 	heta$ will be negative because $\cos 	heta  90^{\circ}$.

Imagine rolling a sheet of paper into a cylinder and placing a bar magnet near its end as shown in the figure. What can you say about the sign of $\vec B \cdot d\vec A$ for every area element $d\vec A$ on the surface of the cylinder?

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