Four point charges are at the corners of a square of side a as shown in the figu

Question

Four point charges are at the corners of a square of side a as shown in the figure below. Determine the magnitude and direction of the resultant electric force on q, with ke, q, and a left in symbolic form. (Let B = 4.5q and C =7.5q. Let the +x-axis be pointing to the right.)

magnitude     direction     Four point charges are at the corners of a square of side a as shown in the figure below. Determine the magnitude and direction of the resultant electric force on q, with ke, q, and a left in symbolic form. (Let B = 4.5q and C =7.5q. Let the +x-axis be pointing to the right.)

Explanation / Answer

Coordinate system:
+x = right
+y = up

Define the leading constant, ke = 1/(4*pi*epsilon0) = 9*10^9 N-m^2/cou^2


Electric force from charge B:
Fbq = ke* Qb*q/a^2, in the direction to the right

As a vector:
Fbq = ke*q * <Qb/a^2, 0>


Electric force the other charge B (upward):
Fbbq = ke*q * <0, Qb/a^2>

Electric force from charge C (upward and rightward):

Total magnitude:
Fcq = ke*q*Qc/( sqrt(a^2 + a^2) )^2

sqrt(a^2 + a^2) = the hypotenuse distance
( sqrt(a^2 + a^2) )^2 = square it for Coulomb's law


Simplify:
Fcq = ke*q*Qc/(2*a^2)

Multiply with the up and to the right unit vector:
Fcq = ke*q*Qc/(2*a^2) * <sqrt(2)/2 , sqrt(2)/2 >


Gather all the formulas for each individual force:
Fbq = ke*q * <Qb/a^2, 0>
Fbbq = ke*q * <0, Qb/a^2>
Fcq = ke*q*Qc/(2*a^2) * <sqrt(2)/2 , sqrt(2)/2 >


Gather all common terms in the leading factor:
Fbq = ke*q/a^2 * <Qb, 0>
Fbbq = ke*q/a^2 * <0, Qb>
Fcq = ke*q/a^2 * <Qc*sqrt(2)/4 , Qc*sqrt(2)/4 >

Add up the x-terms:
Qb + Qc*sqrt(2)/4

Add up the y-terms:
Qb + Qc*sqrt(2)/4

Compose the vector:
Fnet = Fbq + Fbbq + Fcq

Fnet = ke*q/a^2 * <Qb + Qc*sqrt(2)/4, Qb + Qc*sqrt(2)/4>

Combine the vector components in Pythagorean theorem to find magnitude:
| Fnet | = ke*q/a^2 * sqrt((Qb + Qc*sqrt(2)/4)^2 + (Qb + Qc*sqrt(2)/4)^2 )

Simplify:
| Fnet | = ke*q*(Qb + Qc*sqrt(2)/4)*sqrt(2)/a^2

Data:
Qb = 4.5*q
Qc = 7.5*q


| Fnet | = ke*q^2* 4.5 + 7.5*sqrt(2)/4)*sqrt(2)/a^2


Gather:
| Fnet | = (ke*q^2/a^2) *4.5 + 7.5*sqrt(2)/4)*sqrt(2)

Crunch the numeric part of the expression:
4.5 + 7.5*sqrt(2)/4)*sqrt(2) = 8.25


Answer:
| Fnet | = 8.25 * ke*q^2/a^2

You aren't supposed to plug in the values of ke, q, and a, because it tells you to leave them in symbolic form. Even though the value of ke is a universal constant. Be sure to indicate that both q and a are each squared.

45 deg CCW of +x is of course your direction. We know this, because the vector terms are both

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