- #1
Angela G
- 65
- 19
- Homework Statement
- A closed current loop consists of two semicircles with the radius and center at the origin. Den
one is in the yz plane and the other in the xz plane as the figures on the left above show. On a
very large distance R from the origin is a short solenoid coil with N turns, radius b, length
c, center of the point ##( \frac{R}{ \sqrt (2) }, \frac{R}{\sqrt 2}##, 0), and the axis along the diagonal of the xy plane, as the figures shows. We assume that R >> a, R >> b, R >> c
a) Determine the magnetic dipole moments for the loop at the origin and for the solenoid.
b) Determine the mutual inductance of the two loops
- Relevant Equations
- ## \vec m = I \int \vec da ##
##\vec B = \frac{\mu_0 I }{4 \pi} \int \frac{\vec dl \times \vec r}{r^2}##
## \vec A(r) = \frac{\mu_0}{4\pi} \frac{ \vec m \times \vec r}{r^2}##
## \vec B = \nabla \times \vec A ##
## \Phi = M_{12} I_{1 or 2} ##
## \Phi = \vec B \cdot \vec da ##
Hello!
I tried to solve a) see figure below, is it correct?
b) so what I think I can do is to solve ## M_{12} ## from the equation of the magnetic flux then I will get ## \frac{\Phi}{I} = M_{12}## Then I can even use the equation får the magnetic flux and the magnetic field $$ \Phi = \int \vec B \cdot \vec da \Rightarrow \frac{1}{I} \int \vec B \cdot \vec da = M_{12} $$ Now the problem is to determine the magnetic field, I think I can get it either from the Biot-savarts law or the rotation of the vector potential A, since we have the magnetic dipole moment we can determine the vector potential A. I'm stuck here, because I have difficult to dealing with crossproducts and the coordinates. How can I continue? Is there anothe way to get the magnetic field? or the mutual inductance?
I tried to solve a) see figure below, is it correct?
b) so what I think I can do is to solve ## M_{12} ## from the equation of the magnetic flux then I will get ## \frac{\Phi}{I} = M_{12}## Then I can even use the equation får the magnetic flux and the magnetic field $$ \Phi = \int \vec B \cdot \vec da \Rightarrow \frac{1}{I} \int \vec B \cdot \vec da = M_{12} $$ Now the problem is to determine the magnetic field, I think I can get it either from the Biot-savarts law or the rotation of the vector potential A, since we have the magnetic dipole moment we can determine the vector potential A. I'm stuck here, because I have difficult to dealing with crossproducts and the coordinates. How can I continue? Is there anothe way to get the magnetic field? or the mutual inductance?