- #1

- 9

- 1

The problem: a coil of radius

1st solution: solving a differential equation with the current

$$-\frac{\text{d}}{\text{d}y}(B_0N\pi r^2cos(\omega t)+\frac{\mu_0 N^2 \pi r^2 i(t)}{l})=Ri(t)$$

and then the current:

$$i(t)=\frac{Rl^2 N \pi r^2\omega B_0}{((Rl)^2+(\mu_0 N^2 \pi r^2\omega)^2)^2}sin(\omega t)-\frac{\mu_0 N^3(\pi r^2)^2 \omega^2 B_0 l}{((Rl)^2+(\mu_0 N^2 \pi r^2\omega)^2)^2}cos(\omega t)$$

and medium power P:

$$P=\frac{R}{2} \frac{N^2 (\pi r^2)^2 \omega^2 B_0^2 l^2}{(Rl)^2+(\mu_0 N^2 \pi r^2\omega)^2}$$

2nd solution: a virtual ca generator in series with the coil. The circuit impedance is:

$$Z=R+j\omega L$$

the Faraday-Newman-Lentz applied to the solenoid get the max fem:

$$V= N \pi r^2 \omega B_0 sin(\omega t) $$

and the current max:

$$I=\frac{V}{|Z|}$$

so, the medium power dissipated by resistance:

$$P=\frac{R}{2} I^2=\frac{R}{2} \frac{(N \pi r^2 \omega B_0)^2}{R^2+(\omega L)^2}$$

and, substituting ##L=\frac{\mu_0 N^2\pi r^2}{l}##, I obtain the same than 1st solution

3rd soultion: somewhere in internet you can find:

the Faraday-Newman-Lentz applied to the solenoid get the max fem:

$$V= N \pi r^2 \omega B_0 sin(\omega t) $$

so the max current on the resistor:

$$I=\frac{V}{R}$$

and, then the medium power:

$$P=\frac{R}{2} VI=\frac{R}{2} \frac{(N \pi r^2 \omega B_0)^2}{R^2}$$

The question:

1) have you evidence of this calculation?

2) do you know a textbook for this problem?

3) do you know a software to simulate this system (matlab, pspice, labview, ...)?

thamks!

*r*, length*l*and*N*turns, rotating with constant angular velocity*ω*around an axis perpendicular to its simmetric axis and passing for the center of the coil. The coils is submersed in a static magnetic field, intensity*B*, perpendicular to the axis of rotation of the coil. The coil is connected to a resistance R. What is the medium power dissipated by resistance?_{0}1st solution: solving a differential equation with the current

*i(t)*unknow I obtain:$$-\frac{\text{d}}{\text{d}y}(B_0N\pi r^2cos(\omega t)+\frac{\mu_0 N^2 \pi r^2 i(t)}{l})=Ri(t)$$

and then the current:

$$i(t)=\frac{Rl^2 N \pi r^2\omega B_0}{((Rl)^2+(\mu_0 N^2 \pi r^2\omega)^2)^2}sin(\omega t)-\frac{\mu_0 N^3(\pi r^2)^2 \omega^2 B_0 l}{((Rl)^2+(\mu_0 N^2 \pi r^2\omega)^2)^2}cos(\omega t)$$

and medium power P:

$$P=\frac{R}{2} \frac{N^2 (\pi r^2)^2 \omega^2 B_0^2 l^2}{(Rl)^2+(\mu_0 N^2 \pi r^2\omega)^2}$$

2nd solution: a virtual ca generator in series with the coil. The circuit impedance is:

$$Z=R+j\omega L$$

the Faraday-Newman-Lentz applied to the solenoid get the max fem:

$$V= N \pi r^2 \omega B_0 sin(\omega t) $$

and the current max:

$$I=\frac{V}{|Z|}$$

so, the medium power dissipated by resistance:

$$P=\frac{R}{2} I^2=\frac{R}{2} \frac{(N \pi r^2 \omega B_0)^2}{R^2+(\omega L)^2}$$

and, substituting ##L=\frac{\mu_0 N^2\pi r^2}{l}##, I obtain the same than 1st solution

3rd soultion: somewhere in internet you can find:

the Faraday-Newman-Lentz applied to the solenoid get the max fem:

$$V= N \pi r^2 \omega B_0 sin(\omega t) $$

so the max current on the resistor:

$$I=\frac{V}{R}$$

and, then the medium power:

$$P=\frac{R}{2} VI=\frac{R}{2} \frac{(N \pi r^2 \omega B_0)^2}{R^2}$$

The question:

1) have you evidence of this calculation?

2) do you know a textbook for this problem?

3) do you know a software to simulate this system (matlab, pspice, labview, ...)?

thamks!

Last edited by a moderator: