Charge is nonuniformly distributed with linear charge density ? (x)= ? 0 x , whe

Question

Charge is nonuniformly distributed with linear charge density ?(x)= ?0x, where ?0 has units of C/m2, along an horizontal line of length L.

a) Find the total charge on the line. Show your work.
b) Find the electric field at a point P which is on the x axis a distance b from the origin.

(We have gone over uniform charges, but not this stuff and of course being in EP2 it's learn for yourself kind of deal. I am looking for any help on this as it is two lectures into this course and I am beyond lost already.)

Charge is nonuniformly distributed with linear charge density ?(x)= ?0x, where ?0 has units of C/m2, along an horizontal line of length L. Find the total charge on the line. Show your work. Find the electric field at a point P which is on the x axis a distance b from the origin. (We have gone over uniform charges, but not this stuff and of course being in EP2 it's learn for yourself kind of deal. I am looking for any help on this as it is two lectures into this course and I am beyond lost already.)

Explanation / Answer

a) Q = integral of lambda dx = integral of lambda0 x dx from -L/2 to L/2
this is odd
so Q = 0


b) take a little cut on the line, dx
which will have dq = lambda dx = lambda0 x dx

then
dE = k dq/r^2
r = b-x
so E = integral of k lambda0 x dx/(b-x)^2 from -L/2 to L/2

let u = b-x then du = -dx

E = - integral of k lambda0 (b-u) du/u^2 from b-L/2 t0 b+L/2

E = - k lambda0 integral of b/u^2 - 1/u from b-L/2 to b + L/2

E = - k lambda0 ( -b/u - lnu) from b-L/2 to b + L/2

E = k lambda0 (( (b/(b+L/2) + ln(b + L/2)) -(( (b/(b-L/2) + ln(b - L/2))

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