The figure below shows a plastic ring of radius R = 49.0 cm. Two small charged b

Question

The figure below shows a plastic ring of radius R = 49.0 cm. Two small charged beads are on the ring: Bead 1 of charge +2.00 C is fixed in place at the left side; bead 2 of charge +5.50 C can be moved along the ring. The two beads produce a net electric field of magnitude E at the center of the ring. At what positive and negative values of angle should bead 2 be positioned such that E = 2.00 105 N/C? (Measure the angle from the positive x axis taking counterclockwise to be positive.)

Explanation / Answer

q1 = 2.0 uC
q2 = 5.50 uC
E = 2.00 * 10^5 N/C
R = 0.49 m

Net Electric Field, E1 + E2 = 2.0 * 10^5 N/C

E^ = [k*q1/r^2 - k*q2/r^2 * cos()]i^ +  k*q2/r^2 * sin() j^  
|E| = sqrt([k*q1/r^2 - k*q2/r^2 * cos()]^2 + (k*q2/r^2 * sin())^2 ) =  2.0 * 10^5
[(8.9*10^9*2.0*10^-6)/0.49^2 - (8.9*10^9*5.5*10^-6)/0.49^2 * cos()]^2 + ((8.9*10^9*5.5*10^-6)/0.49^2 * sin())^2 ) = (2.0 * 10^5)^2
[74135.8 - 203873.4 * cos()]^2 + [203873.4 * sin()]^2 =  (2.0 * 10^5)^2
74135.8^2 - 2*74135.8*203873.4 * cos() + 203873.4^2 =  (2.0 * 10^5)^2
= 76.5o

positive angle , = 76.5o Counterclockwise from the +x-axis
negative angle , = - 283.5o Counterclockwise from the +x-axis

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