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Coulomb's Law is fundamental in understanding the forces between charged objects. It states that the electrostatic force between two point charges is directly proportional to the product of the magnitudes of their charges and inversely proportional to the square of the distance between them. The formula is given by:\[F = k_e \frac{q_1 q_2}{r^2}\]where:

• \(F\) is the force between the charges,
• \(k_e = 8.99 \times 10^9 \, \mathrm{N\cdot m^2/C^2}\) is Coulomb’s constant,
• \(q_1\) and \(q_2\) are the magnitudes of the charges,
• \(r\) is the distance between the charges.

This law helps us understand that forces are stronger when charges are closer and when their magnitudes are larger. It's important to remember that these forces are vector quantities, meaning they have both magnitude and direction.

In electrostatics, the placement of charges is crucial for determining the forces experienced by these charges. In the given problem, charges are placed along the x-axis:

• A charge \(+q\) at the origin,
• Another identical charge \(+q\) at \(x = 0.50\, \text{m}\),
• A third charge \(+2q\) that needs to be positioned such that its effect is carefully calculated.

For calculations, the relative positioning affects how the forces from each charge interact or add up. In this scenario, the third charge needs to be placed such that its additional force doubles the net force on the origin charge without altering its direction. This requires positioning the \(+2q\) charge somewhere along the positive x-axis but closer to the origin than the \(+q\) charge is.

The net electrostatic force is the total force acting on a charge due to the presence of other charges. It's a vector sum of the individual forces exerted by each charge. In our problem, the initial force is calculated using one charge at \(x = 0.50\, \text{m}\) on the charge at the origin, resulting in:\[F_{\text{origin, initial}} = 4k_e q^2\]To double this force with a third charge \(+2q\), calculations involve setting:\[2 \times F_{\text{origin, initial}} = F_{\text{origin, second}} + F_{\text{origin, third}}\]The purpose of setting this equation is to understand how the third charge contributes to the total force, ensuring the same direction by adding forces algebraically, which in this case both aid in repulsive interactions along the x-axis.

Solving physics problems effectively involves breaking the problem down into manageable parts:

• Understanding the Problem: Clearly identify the knowns and unknowns. Here, we need the position of the third charge so the net force doubles, and its direction remains unchecked.


• Applying Relevant Equations: Use equations like Coulomb's Law to frame the problem mathematically according to what is needed.


• Algebraic Manipulation: Carefully manipulate algebraic equations to isolate variables and solve for unknowns, as seen when determining the third charge's position through solving for \(x\).


• Interpreting Results: Finally, evaluate the solution contextually to see if it makes physical sense, such as the placement of \(+2q\) at \(x = 0.707 \, \text{m}\) to satisfy all conditions of the problem.

This structured approach ensures that each aspect of the physics problem is comprehensively addressed, leading to a correct and meaningful solution.