Understanding Magnetic and Electric Fields: Analyzing Like Poles and Charges

When like poles or charges are brought close to each other, the behavior of the system depends on various factors such as the shape, strength, and matter involved, as well as the external conditions of the experiment. However, for a simpler case, we can model these fields as streams of particles, which makes visualization easier.

Modeling Fields as Streams of Particles

In such a model, the fields can be visualized as streams of minute particles. When like poles or charges are brought close to each other, there will be a force of action and reaction, along with a momentum that depends on the angle of the streams. The alignment and interaction of these streams will determine the resulting forces and energies.

Total Energy in Electric and Magnetic Fields

The total energy in an electric field is given by the energy density integrated over the entire field:

[iiint limits_{text{AllSpace}} frac{1}{2} overrightarrow{E} cdot overrightarrow{D} d^{3}V]

For two like poles, we can denote them with numbers 1 and 2, and the fields due to each as (overrightarrow{E_1}) and (overrightarrow{E_2}) respectively. The total energy density of the resulting field is:

[iiint limits_{text{AllSpace}} frac{1}{2} left(overrightarrow{E_1} cdot overrightarrow{E_2}right) left(overrightarrow{D_1} cdot overrightarrow{D_2}right) d^{3}V]

FLOI and Simplified Energy Calculation

When we FLOI (factorize) the integrand, we get:

[iiint limits_{text{AllSpace}} frac{1}{2} overrightarrow{E_1} cdot overrightarrow{D_1} d^{3}V iiint limits_{text{AllSpace}} frac{1}{2} overrightarrow{E_2} cdot overrightarrow{D_2} d^{3}V - iiint limits_{text{AllSpace}} frac{1}{2} overrightarrow{E_1} cdot overrightarrow{D_2} d^{3}V - iiint limits_{text{AllSpace}} frac{1}{2} overrightarrow{E_2} cdot overrightarrow{D_1} d^{3}V]

The first line of this last expression does not depend on the locations of the poles. It represents the total self-energy of the two poles. If the poles are far apart, so that each pole's field is weak where the other pole's field is strong, then the second line approaches zero because the product of the strengths (factors) will be small in the areas where the fields overlap strongly.

When the poles coincide, the first and second lines become identical. The energy in the field is therefore double the energy at wide separation. Since bringing the poles together increases the energy in the field, there is a force tending to separate the poles.