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Let's delve into the first question:

### Q1: 
Three charged particles lie on the x-axis. The net electrostatic force on particle 3 from particles 1 and 2 is zero, with the condition \( L_{23} = L_{12} \). We need to find the ratio \( q_1/q_2 \).

The condition of equilibrium for particle 3 means the forces due to particles 1 and 2 must cancel each other out. Using Coulomb's Law:

\[
F_{13} = F_{23}
\]

Given \( L_{23} = L_{12} \), we can set up:

\[
k \frac{|q_1|}{L_{13}^2} = k \frac{|q_2|}{L_{23}^2}
\]

Since the distances are equal, we can simplify this to:

\[
\frac{|q_1|}{L_{12}^2} = \frac{|q_2|}{L_{23}^2}
\]

Given that \( L_{12} = L_{23} \), the ratio is:

\[
\frac{q_1}{q_2} = 1
\]

Now, let’s proceed to questions 2 and 3 if you'd like. Would you like me to detail them further?

Three charged particles lie on an x-axis. Particles 1 and 2 are fixed in place. Particle 3 is free to move, but the net electrostatic force on it from particles 1 and 2 happens to be zero. If L23 = L12, what is the ratio q1/q2?

Learn how to calculate the ratio of charges q1/q2 for three particles in equilibrium along the x-axis using Coulomb's Law. Explore the detailed solution with key concepts like electrostatic forces and symmetry conditions in particle systems. Suitable for high school physics students.

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