Apositive charge q1,-45.5 nc is located at the origin. a negative charge q2 =-5.5 nc is located on the positive x-axis, p = 14 cm from the origin.
choose the correct relation to indicate the continuous region on the x-axis that includes a point where the electric field is zero.

 r = 0.1045 cm

Explanation:

The electric field of a point charge is given by the expression

          E = k q / r²

Let's look for the field created by each load

Q₁ charge

           E₁ = k Q₁ / (r-0)²

           E₁ = - 8.99 10⁹ 45.5 10⁻⁹ / r²

           E₁ = - 409.05 / r²

Q₂ charge

           E₂ = k Q₂ / (r-0.14)₂

           E₂ = - 8.99 10⁹ 5.5 10⁻⁹ / (r-0.14)²

           E₂ = -49.45 / (r-0.14)²

The electric field is a vector quantity, so we must use the sum of vectors to get the total field

           E_total = E₁ + E₂

The burden of proof is always considered positive as the two charges are negative the strength is attractive, consider three regions of interest.

- Left of the two charges in this case the two forces are directed to the right and therefore the field is not canceled

- to the right of the two charges, the forces go to the left and the field is not canceled for any distance

- Between the two charges, in this case each force goes in the opposite direction and there is a point for the field to cancel

             0 = E₁ - E₂

             E₁ = E₂

            -409.05 / r² = - 49.45 /(r-0.14)²

              (r-0.14)² = 49.45 / 409.05 r²

             r² - 2 0.14r + 0.14² = 0.12 r²

             r² (1-0.12) - 0.28 r + 0.0196 = 0

             0.88 r² - 0.28 r + 0.0196 = 0

             r² - 0.318 r + 0.0223 = 0

We solve the second degree equation

            r = [0.318 ±√(0.318² - 4 0.0223)] / 2

            r = [0.318 ± 0.109] 2

            r₁ = 0.209 / 2 = 0.1045 m

            r₂ = 0.2135 m

We see that the result in the area of interest between charges is

             r = 0.1045 cm

- There are zones on the sides of the charge, but the force in these zones has a component that takes the test load to the part between the loads, therefore the field is never canceled

whichever one it is, it's definitely not one that you included on the list you provided.

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explanation: