Magnetic Fields
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For the practice problems below, consult your professor for the solution.
The Main Idea
Recall that according to Gauss' law, the electric flux through any closed surface is directly proportional to the net electric charge enclosed by that surface. Given the very direct analogy which exists between an electric charge and a magnetic 'monopoles', we would expect to be able to formulate a second law which states that the magnetic flux through any closed surface is directly proportional to the number of magnetic monopoles enclosed by that surface. But the problem is that magnetic monopoles don't exist. It follows that the equivalent of Gauss' law for magnetic fields reduces to:
[math]\displaystyle{ \Phi_B = \oint B \cdot dA = 0 }[/math]
Realistically, the magnetic flux though any CLOSED surface is zero. The magnetic flux through an area will be its own individual value. This rule is useful when solving for a an unknown magnetic field that's coming from a side of a surface when the other fields from the other sides are known.
Further Description
Gauss's Law for magnetism tells us that magnetic monopoles do not exist. If magnetic monopoles existed, they would be sources and sinks of the magnetic field, and therefore the right-hand side could differ from zero. Gauss's Law for magnetism is one of the four Maxwell's equations, which form the foundation for the entire theory of classical electrodynamics.
The magnitude of the magnetic flux depends on the strength of the magnetic field, the size of the surface area, and the angle between the direction in which the surface area points and the direction of the magnetic field.
Practice Problems
1) There is a small bar magnet with a magnetic dipole D located at the origin (0,0,0). It's aligned with the y-axis. There is a circular disk with a radius of R facing perpendicular to the yz plane and its center is 4 meters away on the +x axis from the bar magnet. What is the magnetic flux going through the disk in terms of the given variable?
2) Referring back to the previous problem, the disk is now tilted so that the angle between the yz plane and the surface is 30 degrees. Find the new magnetic flux in terms of the given variables.
3) A square with side length T is directly facing the xy plane 3 meters away from a current carrying 1 meter wire (from a portion of a nearby circuit powered by a battery with an emf of U). The wire is aligned with the y axis. The magnetic flux going through the square is G. Find the resistance of the wire.