Assume we have an inductor with the following values:
$$B=B_0 \cdot sin(2\cdot \pi \cdot f \cdot t)$$ $$N=10$$ $$r=5cm$$ $$B_0 = 2mT$$ $$f=50Hz$$ $$R=5 \Omega $$ How much current runs through the inductor as function of time?
Well I don't know the inductance, so I tried this:
$$U=-\frac{d \phi}{dt}$$ $$\phi = B \cdot A = 2mT \cdot sin(\pi \cdot 100 \cdot t) \cdot \pi \cdot (2\cdot10^{-2})$$ $$U = -\frac{d}{dt} 2mT \cdot sin(\pi \cdot 100 \cdot t) \cdot \pi \cdot (2\cdot10^{-2})^2 $$ $$U = -\frac{d}{dt} 8 \cdot \pi \cdot 10^{-7}T \cdot sin(\pi \cdot 100 \cdot t) $$ $$U = -8 \cdot \pi \cdot 10^{-7}T \cdot \frac{d}{dt} sin(\pi \cdot 100 \cdot t) $$ $$U = -8 \cdot \pi \cdot 10^{-7}T \cdot 100\cdot \pi \cdot cos(\pi\cdot 100\cdot t) $$ $$U = -800 \cdot \pi^2 \cdot 10^{-7}T \cdot cos(\pi\cdot 100\cdot t) $$ $$I=\frac{U}{R}= - \frac{800 \cdot \pi^2 \cdot 10^{-7}T \cdot cos(\pi\cdot 100\cdot t)}{5\Omega} = -1.579\cdot 10^{-4}\cdot cos(\pi\cdot 100\cdot t)A$$

I never saw something like that anywhere else so I'm a bit unconfident if this approach was really correct. Is this correct? Is there maybe another (easier) approach?

Additional information:
After reading the comments I should add some more information.
As you may have thought, this is an exercise. Here is the full text:
A coil with N turns is brought into a magnetic field. The area of the coil is perpendicular to the field. The coil has a radius r and the field changes after [see B above]. The coil has a resistance of R.
Biot-Savart was discussed but is explicitly not necessary. Solivng this task should be pretty easy, so I get the feeling that this task either lacks some values or I misunderstood it.

  • \$\begingroup\$ \${\displaystyle \Phi _{\mathbf {B} }=LI}\$ You must compute L en.wikipedia.org/wiki/Inductor#Inductance_formulas using B \$\endgroup\$ – Tony Stewart Sunnyskyguy EE75 Jan 27 at 20:06
  • \$\begingroup\$ Thank you! I know this formula. How can I compute L without the length of the coil? As you can see, I know N but not the length. \$\endgroup\$ – TimSch Jan 27 at 20:16
  • \$\begingroup\$ Make assumptions using N,r and R It must be something like AWG40 for R =5 \$\endgroup\$ – Tony Stewart Sunnyskyguy EE75 Jan 27 at 20:17
  • \$\begingroup\$ Just a question: How do you think the Biot-Savart law might apply? Or do you think it would, at all? \$\endgroup\$ – jonk Jan 27 at 23:53
  • \$\begingroup\$ @jonk I added some more information. \$\endgroup\$ – TimSch Jan 29 at 15:20

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