Five charges are arranged as shown in the figure; four identical charges form a square of side d while a fifth charge Q of mass m is suspended above the center of the square, a height d above its plane. (a) What is the charge Q if the charge is in equilibrium? (b) What type of equilibrium does the charge enjoy?

Five charges are arranged as shown in the figure; four identical charges form a square of side d while a fifth charge Q of mass m is suspended above the center of the square, a height d above its plane. (a) What is the charge Q if the charge is in equilibrium? (b) What type of equilibrium does the charge enjoy?

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Question

Five charges are arranged as shown in the figure; four identical charges
form a square of side d while a fifth charge Q of mass m is suspended
above the center of the square, a height d above its plane. (a) What
is the charge Q if the charge is in equilibrium? (b) What type of
equilibrium does the charge enjoy?

Expert Solution

Step 1

For charge Q to be in equilibrium net forces on Q should be 0.

$F_{n e t Q} = 0 F_{n e t Q_{x}} = 0 F_{n e t Q_{y}} = 0$

Net force in x direction on Q

$\begin{array}{rcl} F_{Q_{x}} & = & 0 \\ F_{q_{1}} Q \cos 45 ° + F_{q_{2}} Q \cos 45 ° - F_{q_{3}} Q \cos 45 ° - F_{q_{4}} Q \cos 45 ° & = & 0 \\ (F_{q_{1}} Q + F_{q_{2}} Q) \cos 45 ° & = & (F_{q_{3}} Q + F_{q_{4}} Q) \cos 45 ° \\ F_{q_{1}} Q + F_{q_{2}} Q & = & F_{q_{1}} Q + F_{q_{2}} Q \end{array}$