• However, there is an issue of causality in Physics 1602 2 Electric charge and electrostatic interactions.

    • It contradicts special relativity that all influences propagate instantaneously.
    • We need to confirm the emergence of q from E (blu3mo).
  • So, let’s consider the Electric Vector Field, E(r).

      • Since this only determines E at a single location, there is no issue of causality (?).
    • Fields can be added.
    • It was mentioned that the field can be separated into xyz components.
  • For now, let’s consider the situation of Electrostatics (constant charge function, no charge movement).

    • We want to define the field.
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      • By rearranging this, we can determine E(r).
        • That’s obvious (blu3mo).
  • Understanding the concept of the E field:

    • Two charges, or from a distance, they can be seen as a single consolidated charge.
      • We should understand the method of deriving it using binomial approximation.
    • The imagined situation:
      • There is a charge somewhere on the z-axis, like (0, 0, ±d).
        1. x = 0 - i.e. we evaluate the field on the z-axis.
        • We consider the field on the z-axis.
        1. z = 0 - i.e. we evaluate the field along the x-axis.
        • We consider the field along the x-axis.
        • This behaves quite non-trivially, with the field being maximum at x=±d/√2.
    • The quadrupole field is even more complex.
  • Continuous Distribution:

    • The infinitesimal limit of the discrete case.
    • λ
      • Assuming a uniform charge density, charge / unit length = λ.
    • The line of charge.
      • By separating it into x and y components, we can perform complex integrals (using substitutions, etc.) to calculate it.
      • Important points:
        • If x >> L, it can be considered as a single charge.
        • If x << L, the change is not 1/x^2 but 1/x (?)(blu3mo).
        • This result, using cylindrical symmetry, can be applied to any axis (r-perpendicular).
    • The ring charge.
      • The r-perpendicular term cancels out in the opposite direction, so it can be ignored.
      • In the z-direction:
        • The circumference of the circle, 2πR, appears in the integral.
      • As usual, if z >> R, it changes with 1/z^2.
  • Electric potential:

    • Conservative .
      • It is conservative when the same force is always applied at a certain r.
      • That’s obvious (blu3mo).
    • Conservative E field .
      • Since it is the field instead of the force, the unit differs by C.
      • Therefore, φ is not potential energy!
        • The unit of φ is length * force / charge.
          • This becomes volts, I see~ (blu3mo).
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        • As we did in APMAE2000 Multivariable,
          • ∇Φ+c and ∇Φ are the same.
          • E points in the direction of the greatest change in Φ.
          • Even if the field is 0, the potential may not be 0.
            • That’s obvious, but don’t mistake it for potential being 0 when the forces cancel out, for example.
      • image
        • Simply put, potential is the value of charge multiplied by 1/distance.
          • This can be understood as the derivative being the force.
        • It only depends on distance = There is rotational symmetry.
          • So, calculating the potential of a ring charge, for example, is very simple.
    • I want to read about the ring charge in 4.3.1.
    • Symmetry:
      • Inversion symmetry: When p(x)=p(-x), for example.
        • ;
          • From this, we can see that at Φ(0), .
          • In other words, the force is 0.
      • Reverse inversion symmetry: When p(x)=-p(-x), for example.
        • At x=0, the y and z components of E become 0.
          • I see, it makes sense that the field lines only point left or right in that area.
      • Translational symmetries: When p(x) = p(x+Δx), for example.
        • In this case, p is invariant with respect to x.
          • As a result, Φ and E are also invariant with respect to x.
            • E becomes the gradient of = 0.
        • This is also inversion symmetry.
      • Rotational symmetry:
        • For a ring, it becomes a function of the radial component and the z component.- If the radial component is zero, it means that the radial direction of the electric field is also zero, which can be understood from the concept of inversion symmetry.
  • Ah, I see. This is due to the “infinite set of inversion symmetry.”

  • As a result, all components of the electric field in every direction become zero, so .

  • When we combine this with infinite cylindrical symmetry, we get the following:

    • Rotational symmetry in the direction perpendicular to r.
    • Translational symmetry in the z direction.
  • In the case of a disk charge, the derivative of Φ becomes proportional to |x|.

  • Since the term includes σ = q/πR^2, there are no issues with dimensions.

  • At this point, the gradient (E-Force) becomes constant.

    • Is this what we call a uniform electric field? [blu3mo] [blu3mo]
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