The value of the angle θ for charge Q to be at equilibrium in the given semicircle is determined as 63.42⁰.
Equilibrium of the third charge
The third charge will be in equilibrium position when the force of attraction between Q and Q1 is equal to force of attraction between Q and Q2.
[tex]\frac{kq_1q}{r^2} = \frac{kq_2q}{r^2} \\\\\frac{kq_1q}{(r(180 -\theta))^2} = \frac{kq_2q}{(r\theta)^2}\\\\\frac{q_1}{(r(180 -\theta))^2} = \frac{q_2}{(r\theta)^2}[/tex]
q₁(rθ)² = q₂(r²(180 - θ)²)
q₁r²θ² = q₂(r²(180 - θ)²)
q₁θ² = q₂(180 - θ)²
q₁θ²/q₂ = (180 - θ)²
[tex]\frac{q_1}{q_2} = \frac{(180 - \theta)^2}{\theta^2} \\\\(\frac{q_1}{q_2})^{\frac{1}{2}} = \frac{180 -\theta }{\theta } \\\\(\frac{q_1}{q_2})^{\frac{1}{2}} = \frac{180 }{\theta } - 1\\\\\sqrt{\frac{66.9}{19.8} } = \frac{180 }{\theta } - 1 \\\\1.838 = \frac{180 }{\theta } - 1\\\\2.838 = \frac{180}{\theta} \\\\\theta = \frac{180}{2.838} \\\\\theta = 63.42 \ ^0[/tex]
Thus, the value of the angle θ for charge Q to be at equilibrium in the given semicircle is determined as 63.42⁰.
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