Review Problem. Four identical point charges, each having charge + q, are fixed

Question

Review Problem. Four identical point charges, each having charge + q, are fixed at the corners of a square of side L. A fifth point charge - Q lies a distance z along the line perpendicular to the plane of the square and passing through the center of the square (Fig. P23.65). (a) Show that the force exerted on - Q by the other four charges is F = 4keqQz/(z2 + L2/2)3/2 K Note that this force is directed toward the center of the square whether z is positive (- Q above the square) or negative (- Q below the square). (b) If z is small compared with L, the above expression reduces to F - (constant) zk. Why does this imply that the motion of - Q is simple harmonic, and what would be the period of this motion if the mass of - Q were m? Figure p23.65

Explanation / Answer

z is small compare with L so (z^2+L^2/2)˜L^2/2 so F=-4kqQz/(L^2/2)^3/2 F=-4*2^(3/2)*kqQz/L^3 so the constant is k4*2^(3/2)*Qq/L^3. (let this constant be A) F=-Az=ma=mz'' so mz''+Az=0 this equation implies a simple harmonic motion. =sqrt(A/m)=2/T T=2/sqrt(A/m) Plug A in.

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