Gauss's law and the sum of electric fields we can find that the variation of the force has as response:
- The force is constant and does not depend on the position of the carriages
the electric field is a vector field that indicates the force on a point test charge
Gauss's law is a method to calculate the electric field when the charges have some symmetric
Фl =∫ E. ds = [tex]\frac{q_{int}}{\epsilon_o}[/tex]
The bold letters indicate vectors, fi is the electric field flux, E the field value and ds the area differential of a Gaussian surface.
The Gaussian surface is an imaginary surface at the point where the field is calculated, the shape of this surface should be selected to simplify calculations.
In this case suppose that the plates are very large, so the Gaussian surface is a cylinder with the base parallel to the plane. For this surface the normal is parallel to the field lines so the scalar product is reduced to the algebraic product
2E A = [tex]\frac{q_{int}}{\epsilon_o}[/tex]
The number two is due to the fact that the field is emitted towards both sides of the plate, if we use the concept of surface density
σ = [tex]q_{int}[/tex] / A
[tex]q_{int}[/tex] = σ A
2 E A = σ A / ε₀
E = [tex]\frac{\sigma }{2 \epsilon_o }[/tex]
in the case of the plate with a positive charge the field leaves the plate and in the case of the negative plate the field enters the plate, we can see that the two fields point in the same direction so they must be added
E_ {total} = 2 E
E_ {total} = [tex]\frac{\sigma }{\epsilon_o}[/tex]
The electric force is
F = q E
F = q [tex]\frac{\sigma }{\epsilon_o}[/tex]
From this expression we see that the force is constant and does not change with the position of the cars
In conclusion, using Gauss's law and the sum of electric fields, we find that the answer is:
- The force is constant and does not depend on the position of the carriages
learn more about gauss's law here:
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