Fields

Table of Contents

Equations / Laws

Coulomb's Law

fields_d540e8b0f66dd156eb66bcb7afc00084e1b8fba8.png

where permittivity constant of free space fields_81be2dc60b175dca917d7ee1469fcc0cabceb165.png

Electric Field

fields_12b3fbe48ab13f0fa71e7692f5d2f088d822820a.png

where fields_7a22a4292ffd86592b5af7928035f37f0a1c3e73.png is the test charge, fields_af75e3aee9a5048ccb379313f6cd15dcd718cb1c.png is the electric field at that point.

Magnetic Field

fields_f27bffd707fd710e124298aecc6462568fd02ff5.png

Lorentz Force

This is just writing the magnetic and electric in a single equation.

fields_74182c488b08f7469e5a15687b2b6b512f881b52.png

Electric Flux

fields_4364b69b6c88c4faa88f15ec7a98712a12b1026b.png

Gauss's Law

fields_7d7525e7a2b880abd9df85497ecc9099d4470a9f.png

for Gaussian surfaces, i.e. general closed surface with all surface elements pointing outwards.

Potential Difference as work

fields_55777e1bab715294e01216d4b54169d8369e6bd8.png

I.e. potential-difference is simply work used on a test-charge.

Potential Energy

fields_824da031b06bad80da985ec1bfa653a4edd9eae3.png

Potential

fields_29facb4e8b714f572369346656068876f634e6c7.png

Ampere's Law

fields_5199d7fff4f8220b639e6cdfc0c7198b836c3a63.png

Biot-Savarts

fields_f2db7ddab2dc1f6da2dc9bdc422860dab3bb09c0.png

where fields_4f3f9acf9bd908d9de30ca27981b12e5e19ca671.png is the vector whose magnitude is the length of the differential element of the wire in the direction of conventional current.

Thus, fields_1e64c8df767958f0ce23150bb18a3380979bd93c.png is tangential to the surface of the wire, perpendicular to the current-flow (following the right-hand rule).

Dipole moment

fields_e0b56d1207d63aa1c172cf0e454027c5be283cf7.png

where fields_f4e5566de1c75b7700b562b91a80d2308cc66d12.png is the distance between charges and fields_4212c5de1992b455d07d70aae18d9c7e946b693e.png is the dipole axis.

Continuous Charge Distribution

fields_4203017747f3f1f22a1a7eacf5f0a4d26ccb2047.png

where fields_f3bbd9a34c62cd2cd404756f37bd9047f115b700.png

Torque

fields_8787644fbbb908247c6c9a90b89360613c6c747a.png

Faraday's Law

Induced emf in a conducting loop is given by the rate of change of magnetic flux through the loop, i.e.

fields_a83d4ed2dd3859b0577b4cae729897598a2f9ce5.png

Lenz's Law

The induced current has a direction s.t. the magnetic field due to this current opposes the change in the magnetic field that caused it.

Capacitor

Charge stored

fields_6c324358bedee75e78e88edc6d042207ecc6ebf4.png

Impedence

Definitions

Electric dipole

Two charges of different charge separated by a fixed distance.

Examples

AC

Electromagnetic force fields_224ae917d74dc2133d4403064c971bf562d4db50.png is given by:

fields_d339885b827244190761f08795aba142fbeb6ee5.png

The resulting current is:

fields_a299d6e1b3eec0de8a7dc8ee081ec1857c343b21.png

where fields_d21892f3bdfae9ec08a78f3061484c467c1030ec.png is the phase-difference between fields_224ae917d74dc2133d4403064c971bf562d4db50.png and fields_f78e53d727e82c8e43205bb55f2368f6dc0affa3.png.

How-to

  1. Setup the differential equation for charge wrt. time, using Kirchoff's Law and the fact that the potential difference across all components need to sum to the potential difference across the entire circuit.
  2. Solve said differential equations.

Capacitive load circuit

fields_bf669ea6ffee4be067307f703f9a7a36b42f39c7.png

Thus, "current" across the capacitor fields_10b09a482ede68ce182237efffbfd5cae05a1993.png

fields_600711bf7f9bf5c8a3f66c8f44f8936f34fcce0a.png