# Current induced when dropping a magnet through a coil

When graphing the induced current in a coil while a magnet is dropped through it why is the total area equal to 0? The area represents the charge in the coil but why must the resultant flow of charge be 0?

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## 1 Answer

The EMF induced in the coil is given by:

$$\varepsilon=-N\frac{d\Phi_B}{dt}$$

where $d\Phi_B$ is the magnetic flux through the coil and $N$ is the number of windings. The current through the coil is given by Ohm's law:

$$I=\frac{\varepsilon}{R}=-\frac{N}{R}\frac{d\Phi_B}{dt}$$

where $R$ is the resistance of the coil. The total charge $C$ having gone through a conductor over a period of time is the time integral of the current:

$$C=\int^{t_2}_{t_1}Idt=-\frac{N}{R}\int^{t_2}_{t_1}\frac{d\Phi_B}{dt}dt=-\frac{N}{R}\left(\Phi_B(t_2)-\Phi_B(t_1)\right)$$

Assuming $\Phi_B$ has roughly the same value when the magnet has gone through the coil, $t_2$, as when it was dropped, $t_1$, this will be zero.

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Are those two times, $t_1$ and $t_2$ referring to when the magnet is far above the coil, and far below the coil respectively? – kηives Jan 15 at 23:54
@kηives The formulas are valid for arbitrary times $t_1$ and $t_2$, but I guess they need to be when the magnet is far above and far below for the question to make sense. – jkej Jan 16 at 0:00
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