Consider the collection of electric charges illustrated below surrounding a poin
ID: 1885018 • Letter: C
Question
Consider the collection of electric charges illustrated below surrounding a point of marked with the asterisk, "), where all charges shown are fixed in place and thus 1.) 140 ptsl static: point charge q: (positive): point charge qz (negative): arc of uniformly distributed charge (net charge +Qa, with 1/3 negative and 2/3 positive): line of uniformly distributed charge (net charge -Qa, with 1/3 positive and 2/3 negative): distance from point of interest, Rs; associated angle, 0 distance from point of interest, Ra associated angle, 2 distance from point of interest, R distance from point of interest to the top of the line, Ra (lengths also illustrated) NOTE +20 q: Ry R4 q2 Provide an expression (using the Cartesian system and its unit vectors, x and y, shown) for the electric field, E, caused at the point of interest by the source specified: (k here represents Coulomb's constant) a.) [2.5 pts] electric field, Ei, due to point charge 1 in terms of: k, qu, Ri, 8 b.) 12.5 ptsl electric field, E, due to point charge 2 in terms of: k, qz, R2, 82 c) usptl electric R3, 6, d.) [20ptS] electric field, E, due to the line of charge in terms of: k, aR. field, .due tothe arc of charge in terms ot k, Q.Explanation / Answer
for the givne charge distribution
point charges q1 at R1, 180 - theta
q2 at R2, at R2 , theta2 - 90
q3 at R3 rasdius arc for pi/2 rad
q4 line charge
hence
a. E1 = kq1(cos(theta1)i - sin(theta1)j)/R1^2
b. E2 = kq2(-sin(theta2)i + cos(theta2)j)/R2^2
c. E3 = integral(k*Q3*(cos(theta)i + sin(theta)j)d(theta)/R3^2*theta3) from 0 to theta3 + integral(k*2Q3(-cos(theta)i - sin(theta)j)d(theta)/R3^2*(pi/2 - theta3) )) from theta3 to pi/2
E3 = kQ3(sin(theta3)i - (cos(theta3) - 1)j)/R3^2*theta3 + 2kQ3(-(1 - sin(theta3))i - cos(theta3)j)/R3^2(pi/2 - theta3)
E3 = kQ3[(sin(theta3)i - (cos(theta3) - 1)j)/theta3 + 2kQ3(-(1 - sin(theta3))i - cos(theta3)j)/(pi/2 - theta3))/R3^2
d. E4 = kQ4(La(R4i + La*j/2))/R4*La*sqrt(La^2 + R4^2)*sqrt(La^2/4 + R4^2) - 2kQ4(La/sqrt(La^2 + R4^2) + (La + Lb)/sqrt((La + Lb)^2 + R4^2))(R4i + (La + Lb/2)j)/sqrt(R4^2 + (La + Lb/2)^2)*R4
using formula
E = k*lambda(sin(alpha) + sin(beta))/d
electric field due to finite length
lambda is charge per unit length
alpha and beta are internal angles made by the ling joining the ends of the wire to the poiint of interes
d is vertical distance from the point
E4 = kQ4((R4i + La*j/2))/R4*sqrt(La^2 + R4^2)*sqrt(La^2/4 + R4^2) - 2kQ4(La/sqrt(La^2 + R4^2) + (La + Lb)/sqrt((La + Lb)^2 + R4^2))(R4i + (La + Lb/2)j)/sqrt(R4^2 + (La + Lb/2)^2)*R4