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I am having a hard time understanding the solution of the following question "Draw how the magnetic flux through the coil changes as the coil rotates, starting from the position shown in the diagram". The solution shows a graph similar to the cosine graph which makes me absolutely confused. How can the value of magnetic flux be negative (when the graph is under the x-axis)? According to the definition, magnetic flux is the total number of magnetic field lines that pass through a chosen surface area, so how can that number be negative since after the flux is 0 (when the coil is parallel to the magnetic field lines), at that position the coil can still capture some magnetic field lines? Please give me a simple and elaborate explanation as I am relatively new to the topic. Thank you so much for helping.

this is the question

this is the solution

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The magnetic flux is the amount of field lines (the magnetic flux density) going through a given area. The equation for magnetic flux is $$ \Phi = \vec B \cdot d\vec A = BAcos\theta$$, where both the magnetic flux density and the area are vectors. It is clear mathematically from the dot product that $\Phi$ can be positive (when area and flux are in same relative direction) or negative( when they are in opposite directions).

Your confusion comes from not taking into account that the area we choose must be directed in some way. You might think this is an unnecessary distinction. However, in order for our mathematical formulation of magnetism to be consistent it is important.

Take Faradays law for example. It says that the loop in your diagram will induce an electric current in the direction that opposes the change in magnetic flux. If you claim the flux increases (positively) after the point where it is zero, then what you are saying is that the change in magnetic flux is zero at that point and hence the induced current is zero. Yet, from experiment we know this is not true. There is current and thus we must still be having a change in flux in the negative direction.

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Magnetic flux is a measurement of the total magnetic field which passes through a given area.The measurement of magnetic flux is tied to the particular area chosen.We can choose to make the area any size we want and orient it in any way relative to the magnetic field.

If we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux.The angle at which the field line intersects the area is also important. A field line passing through at a glancing angle will only contribute a small component of the field to the magnetic flux. When calculating the magnetic flux we include only the component of the magnetic field vector which is normal to our test area.

Check this link for more information :Magnetic flux

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I think your confusion lies with believing that flux means number of field lines passing through the area. It is not

Take a look at this wikipedia page https://en.m.wikipedia.org/wiki/Magnetic_flux It clearly defines flux as a surface integral

The field lines way of saying is just a intuitve way to understand flux. Moreover "no. of field lines" is not a physical quantity either. You can draw as many field lines as you want for a particular field(what matters is there relative spacing and such)

So using the surface integral definition and accounting for cosine function in dot product you get yourselves a negative flux value

(Note: while flux is scalar it does not means it cannot have negative value, just that it has no direction,consider scalars like work,charge and so on )

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