Question
We can choose to express the whole system in 2D co-ordinate system by complex notation. Assuming test charge Q at the origin, we can say the position of each q charge is as follows,
Now 13 equal charges, q, are placed at the corners of a regular 13-sided polygon. What is the force on a test charge Q at the center?
Answer
This is a little complicated scenario. There are no pairs of charges sitting at opposite sites. It's a bit delicate to describe it with diagram, we better demonstrate it with calculus.
Each q charge exerts force on test charge Q along the axis defined by the force direction, i.e
Answer
This is a little complicated scenario. There are no pairs of charges sitting at opposite sites. It's a bit delicate to describe it with diagram, we better demonstrate it with calculus.
Each q charge exerts force on test charge Q along the axis defined by the force direction, i.e
$F_{Q_j}$ = Const. $Qq_j $
We can choose to express the whole system in 2D co-ordinate system by complex notation. Assuming test charge Q at the origin, we can say the position of each q charge is as follows,
$q_j - e^{2ij\pi/13} $
Therefore the total force,
$F_Q$ = Const. $ Q \sum_{j=1}^{13} q_j$
considering,
$r = e^{2i\pi/13}$
We can see summation is in a geometric progression. We can write it as (putting r)
$F_Q $ = Const. $ Q \sum_{j=1}^{13}$ $r^j$
$F_Q $ = Const. $ (1-r^{13})/(1-r)$
$F_Q $ = Const. $e^{2i\pi 13/13}$
$F_Q$ = 0
So, the net force experienced by the charge Q is ZERO again.
So, the net force experienced by the charge Q is ZERO again.