Coulomb’s Law for EVs Electric Motor

Coulomb’s Law for EVs Electric Motor – Before we discussed Coulomb’s law, we talked about Lorentz’s law for electric motors. As important as Lorentz’s law is, we will also discuss sufficient material for an electric motor. Here’s the full material.

Coulomb’s Law for EVs Electric Motor

Coulomb's Law for EVs Electric Motor

In its most basic form, Coulomb’s Law describes the force that two charged objects exert on each other (Eq. 5). (Eq. 1- 4, click here.)

    \[\hat{F}=\frac{1}{4\pi\epsilon_0}\frac{q_1q_2\hat{r}}{r^2}\]

As you can see in Eq. (5), the force is proportional to the product of the charges q_1 q_2, so the larger the charges, the stronger the force. It is also inversely proportional to the square of the distance between them, r^2, so the farther away they are, the weaker the force. The direction of the force vector,\hat{F} , is given by the unit vector, \hat{r}, which points from one charge to the other and has a value of 1. If the charges are the same (e.g., both negative), they repel, so \hat{F} is negative, and if the charges are opposite, they attract, so \hat{F} is positive. \epsilon_0 is the “vacuum electric permittivity,” also known as the “electric constant,” which quantifies how well an electric field permeates through a vacuum. Taken together, the fractional first term in Eq. 5 is known as Coulomb’s constant.

The electric field generated by a charged object is expressed as the force per unit charge, newtons per coulomb, or N/C. Therefore, if we divide both sides of Eq. 5 by q_1, we get Eq. 6, which shows the electric field, \hat{E}, generated by q_2:

    \[\hat{E}=\frac{\hat{F}}{q_1}=\frac{1}{4\pi\epsilon_0}\frac{q_2\hat{r}}{r^2}.\]

To make it applicable to more than just a pair of charges, we can rewrite this equation as the differential electric field, dE, due to a differential charge, dq (Eq. 7):

    \[d\hat{E}=\frac{1}{4\pi\epsilon_0}\frac{dq\hat{r}}{r^2}.\]

What does this electric field actually look like? Figure 1 shows the shape and strength of the electric field surrounding a small parcel of charge, dq. The arrows show the direction of the force acting on a charged object with the same kind of charge (if the charges were opposite, the arrows would point inward). The shades of gray in Fig. 1 correspond to the strength of the field: the lighter it is, the stronger the field, and the darker it is, the weaker the field. Note that Fig. 1 only illustrates the field in the two-dimensional plane of the page: in actuality, the field surrounds dq in all directions, and the circles in the diagram actually represent concentric spheres surrounding the charge.

Coulomb's Law for EVs Electric Motor
Figure 1. Electric field surrounding a point charge.

If we want to consider several (or many) charges, the resulting field is simply the sum of all of their individual fields (Eq. 8):

    \[\hat{E}=\frac{1}{4\pi\epsilon_0}\sum_i\frac{dq_i\hat{r}_i}{r_i^2}.\]

where dq_i are the differential charges indexed by the sum, and E is the total field due to all of the charges.

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