Magnetic Fields
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.
Why It Matters
Magnetic flux is interesting because it's always zero through a closed surface because magnetic fields come from dipoles not monopoles where a particle just produces a one-sided magnetic field going in one direction, unlike point charges that generate an electric field going in one direction from every side of the particle. As you read further into this wiki book, you'll find out that the most important factor concerning the production of electricity is magnetic flux. The change of magnetic flux is what creates an electric motor force (voltage) and causes current to flow. The greater the magnetic flux and the faster it changes across a wire or a coil, the more influential it will be in determining the output.
History
Near the end of his career, Michael Faraday proposed that electromagnetic forces can be applied to the empty space around a conductor. This idea was rejected by his fellow scientists, and Faraday did not live to see the day when is proposition was eventually accepted by the scientific community. Faraday's concept of lines of flux generated from charged bodies and magnets provided a way to display electric and magnetic fields in a new way; that conceptual model was crucial for the successful development of 19th century technology.
Mathematician, James Maxwell had an obsession for electricity and magnetism since Faraday's lines of force was read to the Cambridge Philosophical Society in 1855. The paper presented a simplified model of Faraday's ideas and how electricity and magnetism were related. Maxwell took all of that along with what he already knew and developed a linked set of differential equations with 20 equations in 20 variables which will eventually be concatenated into the four equations known as Maxwell's Equations one of which describes magnetic flux as written in the equation displayed above.
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.
See also
Further reading
External links
References
- maxwells-equations.com — An intuitive tutorial of Maxwell's equations.
- Mathematical aspects of Maxwell's equation are discussed on the Dispersive PDE Wiki.