Figure shows variation of Coulomb force (F) acting between two point charges with \( \frac{1}{r^2} \), \( r \) being the separation between the two charges \( (q_1, q_2) \) and \( (q_2, q_3) \). If \( q_2 \) is positive and least in magnitude, then the magnitudes of \( q_1, q_2 \), and \( q_3 \) are such that:
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Remember that the Coulomb force is proportional to the product of the charges and inversely proportional to the square of the distance between them. If one charge is smaller, it produces less force.
Given: The graph displays F (Coulomb force) versus 1/r² for two charge pairs: (q₁,q₂) and (q₂,q₃). Furthermore, q₂ is positive and has the smallest magnitude.
1) Slope-Charge Relationship
The formula for Coulomb force is F = k (q_i q_j) / r². Rearranging for a plot of F versus (1/r²), we get a straight line passing through the origin, where the slope m equals k(q_i q_j).
A positive slope indicates that the product q_i q_j > 0, meaning the charges q_i and q_j have the same sign.
A negative slope indicates that the product q_i q_j < 0, meaning the charges q_i and q_j have opposite signs.
2) Determining Charge Signs from the Graph
The line for (q₁,q₂) shows an upward slope, signifying that q₁ and q₂ share the same sign. Since q₂ > 0, it follows that q₁ > 0.
The line for (q₂,q₃) shows a downward slope, indicating that q₂ and q₃ have opposite signs. Given that q₂ > 0, it must be that q₃ < 0.
3) Comparing Magnitudes via Steepness
The line representing (q₁,q₂) is steeper than the line for (q₂,q₃). Therefore:
|m₁₂| = k|q₁ q₂| > |m₂₃| = k|q₂ q₃| This implies |q₁| > |q₃|, because q₂ is common to both and positive.
4) Incorporating the "q₂ is least" Condition
It is given that |q₂| is the smallest magnitude: thus, |q₂| < |q₁| and |q₂| < |q₃|.
