So I am still a student and I just learned the basics of flux. And my teacher told us that flux ($\phi$) is equal to the magnetic field $B$ $\times$ the area $A$ $\times$ the angle of the area $\cos{\alpha}$. Now I get all this but then comes the induction formula that states that the induction voltage $U_{ind}$ = the number of windings in a coil $N$ $\times$ the derivative of flux against time $d\phi/dt$. Now if I substitute $\phi$ for $AB\cos{\alpha}$ and I know what value is changing, for example $A$, then I can rewrite the formula to $ B\cos{\alpha}\;dA/dt$. But then I look back at the line representation of magnetic fields and think: "If I reduce the area, then the force of the magnetic field $B$ will also decrease some amount?" Now my question is, how much will it change when I shrink the area by, for example, 1 m$^2$? And is there a formula for this? Thank you.
2 Answers
I'm afraid that I can't follow your reasoning, so I don't know if you'll be convinced by these not very profound observations ...
Presumably the magnetic field is due to some source (such as a magnet or a current-carrying coil. The field, that is the magnetic flux density, $\mathbf B$, at each point, will be unaffected by the size or shape of the boundary of the area through which you are calculating the flux. Think of the pattern of lines of magnetic flux; will the pattern be affected by placing in it hoops – of any size or shape – through which flux passes?
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$\begingroup$ Let me try to paint a clearer picture for you: I have a magnet and something of some area A like 1 m^2. If i somehow, without moving or rotating the area, make it smaller to like 0.5 m^2. The "amount" of magnetic field force should be lower. If we look at magnetic fields as tho they had field lines then the amound of field lines should be less when i decrease the area. Hope this clears it up $\endgroup$– Jellyv20Commented Nov 18, 2023 at 19:48
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$\begingroup$ (a) "If i somehow, without moving or rotating the area, make it smaller" But you must shift the boundary of the area. (b) "The "amount" of magnetic field force should be lower." I don't know what you mean by 'magnetic field force'. What is the force acting on? (c) "the amoun[t] of field lines should be less when i decrease the area." Yes, fewer lines will go through the area, but the density of lines and their directions won't change, so the field strength, B won't change at any point. You'll just encircle fewer of the lines. $\endgroup$ Commented Nov 18, 2023 at 21:50
The magnetic field inside a solenoid of surface $A$ and current $I$ is given roughtly by the equation: $$B=\mu_0 \frac{N I}{h},$$ where $h$ is the height of the solenoid. As of this approximation, it does not depend on the surface $A$.
