Things You Need to Know about Electric and Magnetic Fields

(if you want to understand electromagnetic radiation, that is).

This discussion is limited to the physics of electric and magnetic fields needed for an understanding of electromagnetic radiation.

First, think about gravitational field. We define it as gravitational force divided by the mass on which it acts, mg/m. Usually g is meant to be a scalar: the magnitude of the gravitational acceleration vector. When we are talking field, however, g is a vector. Similarly, electric field is the force vector on a charge divided by that charge. Here are the basic ideas you need to understand about electric and magnetic fields.

 1. One way the fields exist in nature is from charges and magnets. Electric field E is on the left and magnetic field B is on the right. They should have the same shape but were not drawn very well:

2. However, we will be concerned with how a changing electric field produces a magnetic field and vice-versa. This is an amazing property of these fields, something that without the experimental evidence, no one would have dreamed up.

E is defined as the (electric) force on a charge divided by that charge. Units: Newton/Coulomb (N/C). B produces a force on a moving charge, perpendicular to both the velocity and B according to the cross product F = qvxB. Therefore the fundamental unit of magnetic field, the Tesla (T), is a
Newton·second/(Coulomb·meter).

 If you do not know about the cross product, vxB is a vector ^ to both with magnitude = vBsinq, where q is the angle between v and B. If you do know, the above is true as well.

3. Flux is field times perpendicular area or perpendicular component of field times area (F_E = EA_┴ or E_┴A and similarly for F_B). Of course if a field varies over the area, one must integrate. Example:
F_B = òB_┴da where da is an area element.

 In case you do not know integration, the above is a fancy way of getting the average B_^ times area.

4. If E is directed upward and it is increasing over time, then ¶E/¶t is also a vector in the upward direction. If E is up & decreasing, then ¶E/¶t points down.

5. Suppose the dots in the picture below represent a rate of change of field (either ¶E/¶t or ¶B/¶t). Then this will cause the existence of the other field circulating around it represented by the circles. If ¶E/¶t = dots then B = circles counterclockwise. If ¶B/¶t = dots then E = circles clockwise. (A dot is a vector directed toward you.) If you do not find this astonishing, then you are not thinking. This is not some technological miracle like the supercomputer; this is a property of nature.

6. The equations for the fields above are

integral of E×dl around a closed loop = -¶F_B/¶t and

integral of B×dl around a closed loop = m_oe_o¶F_E/¶t.
The flux in these equations is the flux through the closed loop of integration. Later we will derive

m_oe_o = c^-2, where c is the speed of light, 299,792,458 m/s » 2.998 x 10⁸ m/s » 3.00 x 10⁸ m/s.

Handling the above equations is simple when the field on the right-hand side of the equation is constant in space (but varies in time). Then the field on the left is constant around a circle, so for example

the integral of E×dl above is E(2pr) and ¶F_B/¶t is pr²¶B/¶t. The signs simply tell the direction. + means follow a right-hand rule. If most people were left-handed or if our thumbs were on the other side, this sign convention would probably be reversed.

7. The energy/volume in E and B are u_E = e_oE²/2 and u_B = B²/(2m_o).

(m_o = 4p x 10^-7 T²m³/J and e_o = 1/(m_oc²))

Armed with this stuff you can now derive the wave nature of light.

Go to one of the main physics pages:
Mechanics

Fluids, heat, electricity and magnetism
Vibrations and waves
Quantum
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