A -10.0 nC point charge and a +20.0 nC point charge are 14.8cm apart on the x-ax

Question

A -10.0 nC point charge and a +20.0 nC point charge are 14.8cm apart on the x-axis. 1. What is the electric potential at the point on the x-axis where the electric field is zero? (V) 2. What is the magnitude of the electric field at the point on the x-axis, between the charges, where the electric potential is zero? (V/m) 3. What is the direction of the electric field at the point on the x-axis, between the charges, where the electric potential is zero? from the positive to the negative charge from the negative to the positive charge up down

Explanation / Answer

If we take the top left particle (or any other, it doesn't really matter - since all the charges are equal and it is a square then the force on each particle due to the other charges will be the same) then the force exerted on it due to the top right charge is: F(tr) = k*q^2 / r^2 - to the left ; where k is 1/(4*Pi*s) I've used s to denote the permittivity of free space, and k is approx = 8.988*10^9Nm^2/c^2. And r = the distance between the charges. Now due to the bottom left charge it is F(bl) = k*q^2 / r^2 - upwards. The resultant of these two forces is F(1) = sqrt( F(tr)^2+F(bl)^2 ) F(1) = sqrt( 2(k*q^2 / r^2 )^2) = sqrt(2)*k*q^2 / r^2 - up/left Now we also have the bottom right charge. If the distance to this charge is R then R = sqrt(r^2+r^2) = sqrt(2)*r so F(br) = k*q^2 / R^2 = k*q^2 / 2r^2 - up/left which is the same direction as F(1) Therefore the overall force on the top teft charge is F = F(1) + F(br) F = [sqrt(2)*k*q^2 / r^2] + [k*q^2 / 2r^2] = [(2sqrt(2)+1)*k*q^2] / 2r^2 This is the same force experienced by all the q-charges due to the other q charges (but the force vectors all point away from the centre). Now for Q the force it excerts of the top left q must be of equal magnitude but in the opposite direction. Since it is at the centre of the square and since the force vectors due to the other q's are pointing away from the centres we know that Q must be negaitive. Since we know this we can simply find the magnitude of Q by: Now F(Q) = F However the distance to Q = 1/2 the distance to the bottom right q; So r(Q) = sqrt(2)*r / 2 k*q*Q / (sqrt(2)*r/2)^2 = [(2sqrt(2)+1)*k*q^2] / 2r^2 Simplifying: 4Q / 2r^2 = (2sqrt(2)+1)*q / 2r^2 Q = [2sqrt(2)+1]*q/4 Q = [2sqrt(2)+1]*(9.29*10^-9)/4 = 8.892*10-9 However we said Q must be negative, so Q = -8.892nC

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