Chapter 27 - Sources of the Magnetic Field

University:
Concordia University Wisconsin
Course:
PHYS 1524 | General Physics II
Academic year:

2021
Views:

364

Pages:

5
Author:

jaxbrokenhart4z4x

Chapter 27 - Sources of the Magnetic Field All our force formulas are inverse square laws ( ). With magnetic fields, additional things need to be taken into account. Gravitational Electric Magnetic Or as an equality For a point mass For a point charge (for an element of current) For a line of current (integration of above) is called the Biot-Savart Law. It is the most general expression for calculating the electric field but is very limited in its practicality. There are only two cases to memorize. Biot-Savart Case 1: Magnetic field from Biot-Savart Case 2: Magnetic field from a curved wire a current ring Because of the geometry, the cross product goes away. Start with the Biot-Savart law. by pythagorean theorem: Substitute Because of the geometry, appears twice, thus Resulting magnetic field is into the page. Since we know the resultant magnetic field is up, we can drop the cross product (since we are accounting for components already). ● Gauss' Law for Magnetism: It's silly!!! Since all magnetic field lines are loops, the flux in will equal the flux out, so thus . This can't really be used to calculate anything. It just serves a reminder of the nature of magnetic flux ( ). ● Ampère's Law - a useful way to calculate magnetic fields, but for a current! ● Instead of integrating around a Gaussian surface, we are now integrating around a Ampèrian loop. We integrate the magnetic field around a closed loop surrounding a current in a plane perpendicular to the current so the magnetic field is always parallel to the Ampèrian loop. For this equation, we have cases. Case 1 - for a straight wire Because of geometry, we can drop the , dot product, and differentials. Case 2 - inside a wire radius distance from the center. Cross section: and In order to find the portion of the current that is enclosed by our Ampèrian loop, we must use a proportion. Case 3 - Inside a solenoid Let's assume that our solenoid has a current and our Ampèrian loop is enclosing number of wires. Putting a rectangular loop cross sectioning the solenoid... doesn't matter because it's outside the coil and don't matter because is perpendicular to note how is a density of wire coils if we call this density , we can simplify this expression to say Case 4 - Inside a toroid Toroid = a wire-wrapped doughnut Current and number of loops Inner radius Integrate across radius where and outer radius Extra note. An old unit for magnetic field: The gauss (G). Conversion factor: Forces due to parallel currents. Parallel currents attract each other. Each wire lies in the electric field caused by the other current. This creates a magnetic force that attracts the two if they're running in the same direction and a magnetic force that repels them if they're running in the opposite direction. this is a force per length, which is practical if the length is unknown or is ideally infinite and thus which looks a ton like coulomb's law and newton's law of gravitation! Use your knowledge of the directions of the currents to determine the direction of the forces.