Question

An unknown positive point charge is placed at x = -0.27 m, y = 0 m, a charge of -2.8 micro-coulombs is placed at x = 0 m, y = -0.20 m, a charge of +2.2 micro-coulombs is placed at x = +0.50 m, y = -0.29 m, and charge of +2.8 micro-coulombs is placed at x = 0 m, y = +0.20 m. If the total electric field due to all four charges at the origin is 4.5 x 106 N/C (the x component of the total electric field is positive), what is value of the magnitude of the unknown charge in micro-coulombs? If negative, include a negative sign.

Answer 1

Given

let  

q1 = q , at (-0.27m,0m)

q2 = +2.8*10^-6C , at (0 m,0.20 m)

q3 = +2.2*10^-6C , at (0.5 m,-0.29 m)

q4 = -2.8*10^-6C , at (0 m,-0.20 m)

the electric field due to a charge q at a point r is  

E = kq/r^2

E1= kq1/r^2 = 9*10^9*q*10^-6/(-0.27)^2 N/ C = q*123456.790 N/C i + 0 N/C j

E2 = E2x i +E2y j = 0 + kq2/r2^2

E2 = 0 i + (9*10^9*2.8*10^-6)/(0.20)^2 N/C = 0 i + 630000 N/C j

E3 = E3x i +E3y j = kq3/r3^2 cos theta i + kq3/r3^2 sin theta j

E3 = -(9*10^9*2.2*10^-6)/(0.29^2+0.5^2)* (0.5/sqrt(0.29^2+0.5^2)) i + (9*10^9*2.2*10^-6)/(0.29^2+0.5^2) (0.29/sqrt(0.29^2+0.5^2)) N/C

= -51264.9432 N/C i + 29733.6671 N/C j

E4 = E4x i +E4y j = 0 + kq4/r4^2  

E4 = 0 N/C i + (9*10^9*(-2.8*10^-6)/(-0.20)^2) N/C j

E4 = 0 N /C i - 630000 N /C

net field is E = Ex i + Ey j

= (E1x+E2x+E3x+E4x)i + (E1y+E2y+E3y+E4y) j

= (q*123456790123.45679+0-51264.9432+0) i +( 0 +630000+29733.6671- 630000)

magntitude is E = sqrt(Ex^2+Ey^2)

4.5*10^6 = sqrt((q*123456.790 -51264.9432)^2+(29733.6671)^2) N/C

solving for q

q = 36.86445 *10^-6 C