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An electric field is a region around a charged object where a force would be exerted on other charged objects. It is quantified by the equation: \[ E = \frac{k \cdot q}{r^2} \] where \( E \) is the electric field, \( k \) is Coulomb's constant \( (8.99 \times 10^9 \, Nm^2/C^2) \), \( q \) is the charge, and \( r \) is the distance from the charge. The direction of the electric field is important as it shows how a positive test charge would move in the field. If multiple charges are present, the total electric field at a point is the vector sum of the electric fields due to each charge individually.
To find where the electric field is zero between two charges, their contributions must exactly cancel each other out.

Point charges refer to charged particles or objects treated as if all their charge is concentrated at a single point in space. This simplifies calculations in electrostatics. For point charges, the electric field can be calculated using Coulomb’s law.
In scenarios like this exercise, point charges are considered to precisely calculate where the electric field is zero. By treating the charges as points, you can use simple algebra and vector addition to solve for fields instead of dealing with the actual size and shape of the charges.

A quadratic equation is a second-order polynomial equation in a single variable \( x \) with the general form: \[ ax^2 + bx + c = 0 \] Solving quadratic equations can be done using several methods, but the quadratic formula is one of the most systematic approaches: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] In this exercise, you rearrange the equation resulting from setting the electric fields equal and then solve the quadratic equation to find the distance \( d \) where the electric field is zero. Every term in the polynomial represents physical quantities, making interpreting the solutions meaningful in a physics context.

Coulomb’s law describes the electrostatic interaction between electrically charged particles. The law states: \[ F = k \frac{|q_1 q_2|}{r^2} \] here \( F \) is the magnitude of the force between the charges, \( k \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the amounts of the charges, and \( r \) is the distance between the charges.
Coulomb’s law is analogous to Newton’s law of gravitation but operates between charges rather than masses. In this problem, it’s used to find the points where the electric fields created by two charges balance out, resulting in a net electric field of zero.